Two Reports on Harmonic Maps

such that E d, = 0. Let t§ = {0 eYi(R3, S2) fl '°°(R3\ U f a,}, S2): deg (0, a) = d,)

where deg (0, a) is the Brouwer degree of ¢ restricted to a small az centred 2-sphere;

f

0, because E d, = 0. Denote by p ..., pQ the list of the a, with d, > 0, with a, repeated d, times; and by n1,..., nQ the list of a, with d, < 0. Let L = mina E, d(p n.,,,), where a is a permutation of the Q indices. Then [Brezis, Coron, Lieb 1, 2] : (3.50)

map in f.

The infimum of the energy of 0 over f equals 47tL, and is not realised by a

(3.51) With respect to the definition off we have (Bethuel, Zheng]: every map in 2,(D3, S2) can be L'-approximated by maps smooth except at a finite number of points.

The space of harmonic maps for negatively curved range (3.52) Suppose that both (M,g) and (N, h) are compact and that 0. Let 1 be a component of 1(M, N) and W the subspace of all harmonic maps in W. Then . is connected [Hartman] and the inclusion map --> % is a homotopy equivalence [Sunada]. In particular, . is an absolute neighbourhood retract of type K(C(n, 1) where 0 e.W and C(ir, 0) is the centraliser of q* tt,(M) in it = n,(N). Also Y.

dim -Y

dim N and .*" is compact.

Also [Schoen, Yau 4], for any xeM the evaluation map ev,.:.

N given by

evz(c5) = O(x) immerses -V? onto a totally geodesic submanifold of (N, h). (3.53) Suppose furthermore that (N, h) is locally symmetric. Then [Sunada] the Riemannian metric induced on .)r by that immersion is independent of the choice of x, and makes Jr a complete locally symmetric space with Riem't < 0. The space ,-l° is also a smooth submanifold of Y'(M, N) (k > m/2) with tangent space T,,(.*') = Ker JJ.

(3.54) [Adachi, Sunada 1].

Suppose (N, h) is a compact Riemannian manifold

89

ANOTHER REPORT ON HARMONIC MAPS

405

with 0. Then there are explicit constants cl > 0 and c2 > 0 depending only on diamTM, diam", volts, vol", lower bounds on RicciM and Ricci' such that

#{,Ye7t°.fa((M,g),(N,h)): E(")

< A2} 5

for any 2.

Here Aa((M, g), (N, h)) is the space of all harmonic maps, and the various .*' its components. Since the energy is constant on a connected family of harmonic maps, we see in particular that the energy spectrum is a discrete subset of R. If moreover 0 somewhere, then for any ,t > 0, the set of surjective harmonic maps 0: (M, g) --* (N, h) with E(0) 5 2 is finite, with cardinality estimated as in (3.54). (3.55)

(3.56) In case M = S', stronger estimates are known (see [Margulis], [Parry, Pollicott], [Adachi, Sunada 2]). (3.57)

Let

0 and for any subspace V c TT(N) set V° = {XeT,(N): = 0 dYe V}.

Let n(y) = max{dim V: V° 0 0}. If 0: (M,g) -+ (N,h) is harmonic and there is a point x e M: rank d¢(x) > n(¢(x)), then no other harmonic map is homotopic to 0. [Adachi, Sunada 2].

4. Maps of Kahler manifolds Complex structures (4.1) Let M be an almost complex manifold; that is, a real manifold with afield J of endomorphisms of T(M) such that J2 = -I. The real dimension of M is even, and we denote it by 2m, calling m the complex dimension. The operator J can be extended linearly to an operator (also denoted by J) on the complexified tangent space Tc(M) (with fibre TT(M) (9 C at x); we denote by T'(M) and T"(M) the bundles of eigenspaces of J on Tc(M) associated to the eigenvalues i and - i. Then

T'(M) (resp. T"(M)) is called the holomorphic (resp. antiholomorphic) tangent bundle;

we have Tc(M) = T'(M) $ T"(M) and T"(M) = T'(M), the complex conjugate.

This decomposition into complex types induces a dual decomposition of T*c(M) = T*'(M) $ T*"(M). We shall suppose M equipped with a Hermitian metric g; that is, a Riemannian metric g such that g(JX, JY) = g(X, Y). The metric g extends to a complex bilinear form on T'(M), and induces on T'M the Hermitian form associating to X, Ye T.' (M) the number g(X, Y). The Kahler form co on (M, g, J) is the 2-form w(X, Y) = g(X, J Y). The manifold (M, g, J) is called almost Kahler if dw = 0.

If the operator J is induced by a complex structure on M (that

is, it is

multiplication by i in the charts of a holomorphic atlas), we speak of complex, Hermitian or Kahler manifolds. Recall that if (Mg) is an oriented surface, it admits a compatible complex structure J such that (M, g, J) is Kahler. Then (M, J) will be called a Riemann surface. On two dimensional domains, conformal invariance of the energy implies that a map 0: (M, g, J) -+ (N, h) is harmonic if and only if it is so for any Hermitian

90

406

J. EELLS AND L. LEMAIRE

metric of (M, J). The conformal class of these metrics will be denoted by Iz, and we shall speak of harmonic maps from the Riemann surface (M, p) to (N, h). (4.2) Let 0: (M, g, J) -+ (N, h, J) be a smooth map between almost Hermitian manifolds. Its complexified differential dcq5 : Tc(M) -+ Tc(N) determines partial differentials by composition with the inclusions of T'(M) and T"(M) in Tc(M) and projections of 7-c(N) on T'(N) and T'(N) as follows:

00: T'(M) -+ T'(N), aq : T"(M) - T'(N);

ac : T'(M) -+ T"(N), a : T"(M) -+ T"(N). We have FO, ac = a O , d co = 00 + ac and d co 1T.(M, = aO + A map 0 is holomorphic if and only if Jo do = do o J, or equivalently 00 = 0; and

antiholomorphic if and only if Jo d¢ = - do o J, or 00 = 0. We say that a map is ±holomorphic if it is holomorphic or antiholomorphic. (4.3)

Define the partial energy densities e'(0) = 1a012 and e"(0) = Q012, and

denote by E'(0) and E"(¢) their integrals. Then E(O) = E'(0) + E"(0), and 0 is holomorphic (resp. antiholomorphic) if and only if E'(0) = 0 (resp. E'(0) = 0). The basic relation between ± holomorphic maps and harmonic maps is the following (see [Report §9] or [Eells, Lemaire 4, §§8 and 9]). Let (M, g, J) and (N, h, J) be almost Hermitian manifolds. If M is cosymplectic (that is, the codifferential of co" vanishes: d*mM = 0) and if N is (1,2)-symplectic (that is, the component of complex type (1, 2) of the 3-form dco' vanishes: 2 = 0), then any ± holomorphic map 0: M - N is harmonic.

Note that these hypotheses are satisfied in particular if M and N are almost Kahler. In that case and if M is compact, E'(¢) - E"(q) is constant on each homotopy class [Lichnerowicz], so that for any variation 0, aE'(O,) at

_ aE"(q,) - 1 aE(O ) - at - 2 at

If (M, g, J) and (N, h, J) are both Kahler manifolds and if r'(0) and T'(0) denote the

Euler-Lagrange operators of E' and E", then T"(¢) = T(¢) and r(q) = r'() +r"(0); consequently r, r' and r' vanish simultaneously. (4.4) An application by [Siu 2] of the unique continuation theorem for harmonic maps shows that if 0: (M, g) --+ (N, h) is a harmonic map between connected Kahler manifolds and if 0 is ± holomorphic on an open set U c M, then 0 is ±holomorphic on M. The same is true if (M, g) is Hermitian cosymplectic and (N, h) almost Hermitian (1, 2)-symplectic. (4.5) ExAMPLE [Fuglede 1]. Let 0 be a ±holomorphic map from a Kahler manifold to a Riemann surface. Then it is a harmonic morphism (with respect to any Hermitian metric on the Riemann surface). Indeed, 0 is harmonic and horizontally conformal.

(4.6) A vector bundle V -+ M is complex if the fibres are complex vector spaces. It is holomorphic if M is a complex manifold and the transition maps are holomorphic (as maps from the intersections of charts to GL (C"). The operator a is well defined on the sections of a holomorphic bundle, because it cancels the transition maps.

91

ANOTHER REPORT ON HARMONIC MAPS

407

If M is a Riemann surface and V -+ M is a complex bundle equipped with a connection, then [Koszul, Malgrange] have shown that V admits a holomorphic bundle structure such that a/az = Vale, on sections.

The idea of the proof is to find local frame fields b = (b,... b,) such that

Vale, b5 = 0. Then the transition maps are holomorphic. The existence of such frames follows from a local existence theorem for complex differential systems. (Of course, this is a form of integrability of an almost complex structure-the special case in one

complex dimension of the Newlander-Nirenberg theorem.) (4.7)

As an application (first made by [J. C. Wood 1]), we see that if

0: (M, g) -> (N, h) is a harmonic map from a Riemann surface to a Kahler manifold, then

the complex bundle q-'T'(N) -+ M admits a holomorphic structure such that 00 is a holomorphic section of the holomorphic bundle T'*(M) 0 c-'T'(N). Indeed, az

a`f' = Va/af aq5 = 0.

(4.8) Let 0: (M, g) - (N, h) be a harmonic non ±holomorphic map from a Riemann surface to a Kahler manifold. Then 0 is ±holomorphic on at most a discrete set.

That is because 00 and a0 are holomorphic sections of the appropriate bundles, and therefore vanish only at isolated points.

Consider now a holomorphic vector bundle V over the sphere S2 (for example, a complex bundle with its Koszul-Malgrange structure). By the Birkhoff-Grothendieck structure theorem [Grothendieck], V decomposes into a direct sum of holomorphic line bundles (4.9)

V=L,BL2(D ...®Lk. The isomorphism classes of the L, are uniquely determined; they can be ordered so that c,(L,)

c,(L2) > ... > c,(Lk),

where c,(L) denotes the first Chern class of L. We remark that for a line bundle L over S2 the Riemann-Roch theorem implies that the dimension of the space of holomorphic sections is

c,(L)+1

if c,(L) >, 0,

0

if c,(L) < 0.

Curvature restrictions and rigidity We now turn to the first main applications [Siu 2, 3, 4] of the theory of harmonic maps to Kahler manifolds. We first introduce various notions of negative curvature. For detailed expositions see [Siu 5] or [Wu]. (4.10)

Let (N, h) be a Kahler manifold, and define a Hermitian form Q: (T ,(N) (9 T;(N)) x (T,,(N) (& Tv(N)) --+ C

92

408

J. EELLS AND L. LEMAIRE

as follows. For any X, Y, Z, We T' (N), set

Q(X(9 Y,Z® W) = (R(X,Z) W, Y> where R is the curvature tensor of (N, h), and the definition is extended to arbitrary elements of T ,(N) ® T'(N) by requiring linearity in the first factor and conjugate linearity in the second. Call decomposable any element in T' (N) ® T'(N) of the form Z ® W. Say that Q is negative definite (resp. semi-definite) of level k on N if Q(A, A) < 0 (resp. < 0) for

each non-zero element A e T' (N) ® T'(N) which can be expressed as the sum of at most k decomposable elements. Following the original terminology of [Siu 2, 5], say that the curvature tensor of a Kahler manifold is strongly negative (strongly semi-negative) if Q is negative definite (resp. semi-definite) of level 2 at every point; and is very strongly negative (resp. very

strongly semi-negative) if Q is negative definite (resp. semi-definite) at all levels at every point. We remark that a Kahler manifold with strongly negative curvature tensor has negative sectional curvature, and that it has negative holomorphic bisectional curvature (4.24) if and only if Q is negative definite of level 1. (4.11) ExAMPLES. The open disc D" c C" with its Bergman metric has very strongly negative curvature [Siu 2] ; the product D' x ... x D' has strongly seminegative curvature [Jost, Yau 1, 2]. In both cases, quotients by discrete groups of holomorphic isometries provide examples of compact manifolds with the same properties. On the other hand, [Mostow, Siu] have given examples of compact manifolds with very strongly negative curvature which are not quotients of D" by

discrete groups. (4.12) [Siu 2]. Let (M, g) and (N, h) be compact Kahler manifolds with RIN, I) strongly semi-negative. If 0: (Mg) -+ (N, h) is a harmonic map and there is a point at which k'-') is strongly negative and rank, 0 >, 4, then 0 is ±holomorphic. As a consequence, [Siu 2] also shows that if dime M > 2 and is strongly negative, then any harmonic oriented homotopy equivalence 0: (M, g) --> (N, h) is a biholomorphic diffeomorphism.

(4.13) Now if N is compact and 0, every homotopy class of maps contains a harmonic representative (3.33); therefore the above results led [Siu 2] to the existence of holomorphic maps and to the following : STRONG RIGIDITY THEOREM. Let (M, g) and (N, h) be compact Kahler manifolds of complex dimension >,2, having the same homotopy type. If is strongly negative, then M and N are ± biholomorphically equivalent. As is usual with such assertions, it suffices in (4.12) and (4.13) to assume R("? )

strongly semi-negative, and strictly so somewhere; see [Wu].

The same conclusion is valid if (N, h) is a compact quotient of an irreducible Hermitian symmetric space of non-compact type. A refinement of the proof is needed, because its curvature is only strongly semi-negative ([Siu 2]). (4.14) The deformation theory of Riemann surfaces (of positive genera) shows that there is no analogue of (4.13) if dime M = 1.

93

409

ANOTHER REPORT ON HARMONIC MAPS

(4.15)

To start the proof Siu makes a modification of the Weitzenbock formula [Report (3.13)] giving an expression for the Laplacian of the energy density of a harmonic map. In general, that involves the curvature of both domain and range. In the present version, however, only the curvature of (N, h) appears. Siu computes as < h, a0 A

a

a are the components of the exterior differential applied

to forms on M. For any smooth map,

as =

acbAVaa >

where, for example, V is the (1, 0)-component of the covariant exterior derivative of the form Do with values in 0-1T'N. Taking the exterior product of the above 4-forms with (coM)ii-2 and integrating yields a formula in which all terms have the same sign.

Computations involving the curvature condition show that ao or a¢ is zero in a neighbourhood of a point. The result then follows from (4.4). See also [Siu 5] for a discussion of various related Weitzenbock formulas and their application to similar situations. (4.16) In very broad terms, the proof of the strong rigidity theorem (4.13) combines an existence theorem for harmonic maps with a second order formula

which introduces a curvature-related restriction. This pattern is at the basis of various applications of harmonic maps to the study of Riemannian or Kahler manifolds (see (4.29), (6.23), (6.33)). However, we shall see that these results do not follow from a simple repetition of that scheme. be strongly negative. If a homology class in (4.17) APPLICATION [Siu 2]. Let H2k(N, Z) can be represented by the continuous image of a compact Kahler manifold of

complex dimension 3 2, then it can be represented by a complex analytic subvariety of N. (4.18) For any map 0: (M, g) - (N, h) between Kahler manifolds, we construct >J = (5*h)2,0. If 0 is ±holomorphic, then q 0. Recall [Report (10.5)] that if M is a

Riemann surface and 0 a harmonic map, then I is a holomorphic quadratic differential (that being true for any Riemannian manifold (N, h)). (Sampson 2, 3, 4] provided a simplified proof of Siu's result (4.12) by showing that if (M, g) and (N, h) are compact Kahler and Rlr',') strongly semi-negative, then for any harmonic map (M, g) --> (N, h), the form rl is a holomorphic (2, 0)-tensor field on M. (4.19)

The following result is due to [Sunada]. Let M be a compact Kahler

manifold and N = D/I' a free quotient of a bounded symmetric domain by a discrete group of holomorphic isometries. If two holomorphic mappings 00, q,: (M, g) -- N are homotopic, then their lifts to the universal covering differ by a holomorphic automorphism of D.

The methods and results of Siu have been extended to various other situations. For instance, combining work of [Jost, Yau 1, 2] and [Mok 1] gives the (4.20)

following. Let (M, g) be a compact Kahler manifold of complex dimension ? 2. Let (IV, h) = 9, x ... x Nk be a Hermitian symmetric space of non-compact type, decomposed into irreducible factors. Let (N, h) be a compact finitely irreducible quotient of (N, h) (that is, (N, h) has no finite Riemannian cover by a Riemannian product), and diffeomorphism 0: M -+ N suppose that N has the homotopy type of M. Then there is a

94

410

J. EELLS AND L. LEMAIRE

k: tll - N, x ... x Nk has ± holo-

whose lift to the universal cover morphic components.

This assertion should be compared to the complex form of Mostow's rigidity theorem on symmetric complex structures [Mostow]. A new ingredient in the proof is the study of the foliation defined by the level sets of the maps fit.

See also [Jost, Yau 3] for an extension to non-compact manifolds of finite volume. (4.21) [Jost, Yau 1] also apply their method to show that every deformation (in the sense of Kodaira-Spencer) of a Kodaira surface is again a Kodaira surface. In other words, the moduli space of a Kodaira surface is that of the Riemann surface on which it is fibred. (4.22) Let M be a compact submanifold of dimension >, 2 of a compact Kahler manifold (N, h) with R"M'" strongly negative [Kalka], or of a compact irreducible quotient of a polydisc [Mok 1]. Then the deformations of M as a submanifold of N coincide with those of M as a complex manifold.

Second complex variation (4.23) Let 0: (M, g) - (N, h) be a harmonic map from a compact Riemann surface to a Kahler manifold. Consider a variation of 0 with complex parameter

s e C. As in (4.3) we have

aE'(q) _ OE' (0) as

as

and

2 a2E"

as as

(_)

as as

(4.24) The following expression is due to [Siu, Yau 2] (and [Suzuki 1, 2] for a special case) : C92 E'

as as ,8-0

=

( IV8 fm

+


I2+IV 5 2+
T,) 00, ¢Z> - 2Re ) dz A dz

where <,> denotes the bilinear extension of h to Tc(N).

Note that = - 41Re 0812 IRe 0,12

HBRiem"((l8, 0)

where HBRiem' denotes the holomorphic bisectional curvature of N: HBRiem'(X, Y) = . (4.25) Suppose now that for its Koszul-Malgrange complex structure (4.6), the bundle 0-1T(N) --, M admits a non-zero holomorphic section v. Then a variation 0: C x M -+ N of 0 can be constructed such that a0 as

8.0

=0,00 as

=v. 8.0

95

ANOTHER REPORT ON HARMONIC MAPS

411

Consequently (4.24) reduces to a2E"(q) as as

=-4JMJRe¢82IRe¢,2 8-0

We are led to the following result of [Siu, Yau 2] : (4.26)

Let 0: (M, g) -> (N, h) be a harmonic non ± holomorphic map from a compact Riemann surface to a Kahler manifold. If HBRiem",°) > 0 and if there is a non-trivial holomorphic variation ve '(O-1T(N)), then E-index 0 > 0. Indeed, the above construction based on the variation v gives a2E"(qs) os os

8-0

and provides a real variation of 0 along which the second derivative of E is negative. (4.27)

An application of the Riemann-Roch theorem to the holomorphic vector bundle 0-1T(N) -; M (with its Koszul-Malgrange structure) ensures the existence of holomorphic variations in various situations. For instance [Eells, Wood 4]: Let (M,µ) be a compact Riemann surface of genus M = p and (N, h) a Kahler manifold with HBRiem"', " > 0. If 0: M --> (N, h) is harmonic and not ±holomorphic, then

E-index (0) >, (0*c1(N))[M]+n(1-p), where c1(N) is the first Chern class of T(N). (4.28)

If ¢ is a map from a compact oriented surface M to a manifold N such

that H2(N, Z) = 7L, the degree deg (0) of 0 is the image of the generator of H2(N, 7L)

in H2(M, Z) = Z by the homomorphism induced by 0. EXAMPLE.

If (N, h) =

and deg (0) > 0, then

E-index (0) > deg (q5) (n + 1) +n(1-p).

In particular, take n = 1 and p >, 1. Then for any integer d with Idl <, p there is [Lemaire 3] a compact Riemann surface M of genus M = p and a harmonic non ±holomorphic map 0: M -- S2 = P1(C) of degree d. For d >_ p/2, it must have positive index. (4.29)

[Siu, Yau 2] use (4.26) to give a proof of a conjecture of Andreotti-

Frankel : any compact Kahler manifold N with HBRiem" > 0 is biholomorphically equivalent to the complex projective space Very briefly, the proof goes as follows. Earlier considerations had shown that (a) N could be assumed simply connected and H2(N; Z) = Z; and (b) the problem could

be reduced to showing that a generator of the free part of H2(N,Z) could be represented by a rational curve, that is, a holomorphic map from S2 to N. [Siu, Yau 2] apply a theorem of [Sacks, Uhlenbeck 1] (see (5.3) below) ensuring that a set of generators of ir2(N) = H2(N, Z) can be represented by energy minimising harmonic maps 0, of S2. Using the Birkhoff-Grothendieck decomposition of the bundle OT 'TN and a computation of its Chern class, they establish the existence of a holomorphic

96

412

J. EELLS AND L. LEMAIRE

variation of qf, so that by (4.26) each 0, is +holomorphic. Finally, the theory of deformations of curves is used to show that a given generator is represented by a single holomorphic map of S2.

Another proof of (4.29) in a more general context, based on the methods of algebraic geometry, was obtained at about the same time by [Mori]. [Mok 2] settled a conjecture of Siu and Yau as follows. Let (N, h) be a compact Kahler manifold with HBRiem""" > 0. Then its universal cover (N, h) is biholomorphically equivalent to C'` x No for some k and some compact Hermitian symmetric space No. In fact, he proves that there are non-negative integers k, n,, ... , nq and compact irreducible Hermitian symmetric spaces N1, ... , N, of rank >,2 such that (N, h) is isometrically biholomorphic to (4.30)

C' x Ni x ... x NP X (P,,,(C), B,) x ... x

0),

where 0t is a Kahler metric on P,,,(C) with HBRiem > 0.

Previous work in this direction was done by [Howard, Smyth] in case n = 2, [Bando] for n = 3 and by [Siu 3, 4] and [Mok, Zhong]. An important idea in Mok's proof (due to [Bando] in case dimR N = 3 and to [Hamilton 2, 3] in a related situation for dimR N = 4) is the following. If (h1) is a solution of Hamilton's heat flow for Hermitian metrics, At+Ricci(ht) = 0;

0 for all t > 0. If Ricci (ho) > 0, then 0, then Ricci (h1) > 0 for all t > O. Mok then uses a splitting theorem of [Howard, Smyth, Wu] to reduce (4.30) to the

and if

proof of: (4.31) Let (N, h) be a compact irreducible Kdhler manifold with HBRiem(",'`' > 0 0 somewhere. Then either (N, h) is isometrically biholomorphic to a and Hermitian symmetric space of rank > 2 or (N, h) is biholomorphic to This was proved by [Cao, Chow] under the stronger hypothesis that (N, h) is simply connected and its curvature operator (see (6.28) below) is non-negative. Mok's proof of (4.31) uses the algebraic geometric ideas of [Mori]. (4.32)

Let f be a real valued function on the Kahler manifold (M, g) and let

1 < q < m = dime M. f is called q-plurisubharmonic (resp. strictly so) at x if the sum of any q eigenvalues of V V f(x) is >0 (resp. >0). A relatively compact smooth domain Mo in (M, g) is said to have q-hyperconvex

boundary (resp. strictly q-hyperconvex boundary) if Mo = f f < 0} where f is qplurisubharmonic (resp. strictly so) and JVfI = 1 along BMo. A function f is subharmonic if it is m-plurisubharmonic; and is plurisubharmonic if it is 1-plurisubharmonic. (4.33) The next result is due to [Siu 5]. Let (M, g) be a Kahler manifold with HBRiem(M,°' > 0, and Mo a smooth relatively compact domain in M with strictly q-hyperconvex boundary, for some q with 1 < q < m. Then iti(M, M\M(,) = 0 for

i < m-q.

97

ANOTHER REPORT ON HARMONIC MAPS

413

It follows that if M is compact and B a complex submanifold having a tubular neighbourhood V with smooth boundary such that a(M\V) is strictly q-hyperconvex

(for some 1 < q < m), then nt(M, B) = 0 for i < m-q. The proof is based on the existence of a strictly q-plurisubharmonic exhaustion function on M0, and on Morse theory. Holomorphicity of E-minima (4.34)

There are other situations in which an energy minimising map is

holomorphic. Indeed [Eells, Wood 2]: Let M be a compact Riemann surface and (N, h) a simply-connected almost Kahler manifold with n2(N)=Z holomorphically generated. Then any E-minimising harmonic map 0: is ±holomorphic. See (6.47) for examples of such (N, h). Some homotopy hypothesis is needed, for there are simply connected K3 surfaces N for which every holomorphic map 0: M -+ N is constant [Mayer], and yet n2(N) is generated by E-minimising harmonic maps by (5.3) below. There are also complex tori without holomorphic curves (Siegel).

3), then its Gauss APPLICATION. If 0: M -* T" is a conformal immersion (n map y, is an E-minimum if and only if either M is a Euclidean sphere in 183 or 0 is a harmonic immersion. (4.35) The holomorphic sectional curvature of a Hermitian manifold at a point x E M in the unit direction X e Tx (M) is
4c

(deg (0) - 2(p - 1)),

then 0 is ±holomorphic. Using this, [Udagawa 2] proved that if 0 is a harmonic isometric immersion of (S2,g) in P (C) of degree 0, then vol (S2,g) > l6n/c, equality holding if and only if (S2, g) is geodesic.

the Euclidean sphere of radius c/4 and ¢ is totally real and totally

Rank restrictions (4.36) [Sampson 4] modified his technique in (4.18) to draw these conclusions: Let (M, g) be a compact Kahler manifold and (N, h) a Riemannian manifold for which <, 0 for all X, Ye Tc(N). If 0: (M, g) -+ (N, h) is harmonic, then I = (O*h)2'0 is a holomorphic quadratic differential on (M,g). Furthermore, V 00 = 0, Y) Y, X> = 0 for all X, Ye dc(T'(M)). and As a consequence, the harmonicity of 0 is independent of which Kahler metric is given on the complex manifold M. (4.37) Let (M, g) be a compact Kdhler manifold and (N, h) a locally symmetric space of non-compact type. Then [Sampson 4], [Carlson, Toledo] : a map 0: (M, g) --> N is harmonic if and only if Vz is the a-operator of a holomorphic structure on 0-'Tc(N)

and 00 is a holomorphic section of T'*(M) ® 0-'T(N).

98

414

J. EELLS AND L. LEMAIRE

(4.38) [Carlson, Toledo].

If (M, g) is a compact Kdhler manifold and (N, h) a

locally symmetric space of non-compact type which is not locally Hermitian symmetric,

then any harmonic map 0: (M, g) --' (N, h) has rank 0 < n at every point. The proof uses (4.37) and a detailed study of the Lie algebra associated to (N, h), in which d¢(T'-0(M)) appears as an Abelian subspace. Maps to space forms (4.39) Denote by RN(c) a real space form of constant sectional curvature c, by CN(c) a complex space form of constant holomorphic curvature c, and by HN(c) the analogous quaternionic space form. We have the following restrictions on harmonic maps of Kahler manifolds. (4.40) The curvature of a complex space form with c < 0 is strongly negative. Thus, Siu's theorem (4.12) implies that any harmonic map of a compact Kdhler

manifold (M, g) to CN(c) with c < 0 and rankR 0 > 4 somewhere is ±holomorphic.

Without assuming that M is compact, the same conclusion was drawn by [Dajczer, Thorgbergsson] and [Udagawa 1] provided 0 is a harmonic isometric immersion.

An easy consequence is the following theorem of [Dajczer, Rodriguez]. With c < 0 and (M, g) Kdhler (m > 2), there is no minimal isometric immersion of (M, g) into RN(c). (4.41) Let (Mg) be compact Kahler and c < 0. Then [Sampson 31: any harmonic map 0: (M, g) --> RN(c) has rankR do < 2 on M. Also [Carlson, Toledo] : If 0: (M, g)

-+ CN(c) is a harmonic map having rankR d¢ - 3 at some point of M, then 0 is ±holomorphic. If ¢ : (Mg) -+ HN(c) is harmonic, then rankR do < dimR HN(c)/2 on M. (4.42) Carlson and Toledo reformulate the main ideas leading to (4.40) in a twistorial manner - in the spirit and terminology of Section 7 below. Set

D = HN(-1) = Sp(n, 1)/Sp (n) x Sp (l), and form the indicated homogeneous fibrations it with fibre S2. Here r denotes a discrete subgroup of Sp (n, 1), acting freely and isometrically on both D and D :

_

Sp(n, 1)

D

Sp(n) x U(1)

D

Sp(n, 1) Sp(n) x Sp(l)

D=N D = N.

Now D is a homogeneous complex manifold. They prove : (a) If q : (M, g) -+ N is horizontal and holomorphic, then n o 0 _ 0: M - N is harmonic. (b) If 0 : (M, g) --+ N is harmonic with rankR do 3 at some point of M, then there is a horizontal holomorphic lift : (M, g) -* N. In case dim, M = 1, the twistor fibration it was used by [Glazebrook 1].

99

ANOTHER REPORT ON HARMONIC MAPS

415

(1,1)-geodesic maps (4.43) If (M, g) is almost Hermitian and (N, h) Riemannian, say that a map 0: (M, g) -> (N, h) is (1, 1)-geodesic if (Vd0)',1 = 0, the superscripts indicating the component of complex type (1, 1) of the 2-tensor.

Any (1, 1)-geodesic map is harmonic ; and conversely, if dime M = 1. When 0 is (1, 1)-geodesic, the quadratic differential r/ = (0*h)2'0 is holomorphic

(4.44)

[Ohnita 2]. A map 0: (M, g) -> (N, h) of a (1, 2)-symplectic manifold is (1,1)-geodesic if and only if its restriction to every complex curve in M is harmonic [Rawnsley 2].

Let 0: (M, g)

(N, h) be a map between Kahler manifolds, and V,00 the

composition of (Vd0)1"' with the projection on T'(N). Then 0 is (1, 1)-geodesic if and only if Vaa0 = 0 or equivalently V?a0 = 0. In this case, the notion of (1,1)-geodesic

map does not depend on the choice of the Kahler metric g on M. Recall that 0 is harmonic if and only if r'(0) = Trace V. a0 = 0. Finally observe that a ±holomorphic map between (1, 2)-symplectic manifolds is (1, 1)-geodesic, so that (1, 1)-geodesic maps lie between harmonic and ±holomorphic maps.

See Items Added in Proof, p. 501. As an application of the holomorphicity of r/, Ohnita and Rawnsley have shown that if (M, g) is a simple (that is, with simple isometry group) Hermitian symmetric (4.45)

space of compact type, then any (1, 1)-geodesic map 0: (M, g) --* (N, h) which is nowhere an immersion must be constant. (4.46) [Dajczer, Rodriguez]. If 0: (M, g) - RN(O) is a minimal isometric immersion of a Kahler manifold into a flat manifold, then 0 is (1,1)-geodesic. (4.47) The next results are due to [Udagawa 2], following special cases for c > 0 of [Dajczer, Gromoll], [Dajczer, Thorbergsson], [Udagawa 1] and [Rawnsley 2]. Let 0: (M, g) -> CN(c) be a (1, 1)-geodesic map of a Kdhler manifold to a complex space form with c 0 0. If ranks >, 3 at some point of M, then 0 is ± holomorphic. If 0 maps into a real space form l N(c), then rankR 0 < 2 on M.

CR-maps (4.48) An almost CR (Cauchy-Riemann) structure on a Riemannian manifold (N, h) is a real 1-codimensional subbundle H c T(N) with a complex structure J' (that is, a bundle endomorphism of H with (J")2 = - Id); it is CR-Hermitian if

hy(u, v) = h,(J"u, J"v) for all u, v e Hy and y c- N; and CR-Kahler if the Kahler form w, defined by w(u, v) = h(u, J"v) for u, v E H and w(u, v) = 0 for u 1 H, satisfies dw = 0.

Note that a CR-manifold is odd dimensional.

A map 0: (M, g, J) - (N, h, J") of a complex Hermitian manifold to a CRHermitian manifold is called a CR-map if d0(T'(M)) c H and d0 o JTM = J" o d0. (4.49) [Burstall 2] has shown that if 01, 02 are two CR-maps which agree on an open subset of M, then 0, = 02 on M.

100

416

J. EELLS AND L. LEMAIRE

(4.50) An almost CR-structure is integrable if the space of sections of the bundle H' (that is, of type (1, 0)) is closed under the Lie bracket. For instance, any almost CRstructure on a 3-manifold is integrable. EXAMPLE. If (N, h) is a real hypersurface of a Kahler manifold (N, h, I), then (N,h) is CR-Kahler and integrable, with bundle H = T(N) A JT(N) and J = Jl,,.

ExAMPLE. Let 0: (N, h) - (9,.T, h) be a Riemannian circle bundle over a Kahler manifold. Define a complex structure J" on the horizontal distribution H by taking for JHX the horizontal lift of Jn,,(X), for all X e H. Then (N, h, JH) is almost CRKahler, but not integrable in general.

(4.51) [Burstall 2], [Rawnsley 2]. If (M, g, J") is almost Hermitian cosymplectic and (N, h, JH) is almost CR-Kahler, then any CR-map 0: M -* N is an E-minimum (and so harmonic). (4.52) [Burstall 2] has also produced conditions ensuring that a map between CR-manifolds be harmonic.

5. Maps of surfaces (5.1) In this Section, (Mg) will always denote a compact Riemannian surface, not necessarily orientable, and (M, p) a compact Riemann surface (with a conformal class of metrics p). Recall (4.1) that for any Riemannian manifold (N, h), harmonic maps 0: (M, y) - (N, h) are well defined as the harmonic maps 0: (M, g) (N, h) for any g in the class p.

Existence of harmonic maps of surfaces (5.2) As we have seen in (3.34), Morrey's regularity theorem and the control of n,-action imply that if N is a compact manifold with 7t2(N) = 0, then any homotopy class of maps from (M, t) to (N, h) contains an energy minimising harmonic map. (5.3)

Recall (see [Report, (10.14)-.(10.17)J) that using the perturbed energy (1 +Idol 2)"dx

EJO) = JJJ

(a > 1),

M

[Sacks, Uhlenbeck 1] have shown that there exists a set of components *'r of r'(S2, N), each containing a minimising harmonic map and whose orbits under the action of n,(N) contain generators of nc2(N).

Putting aside the difficult analysis, here is a rough idea of the proof. In a given homotopy class .3° of maps from S2 to N, the perturbed energy E. (a > 1) attains its minimum at a map 0a. When or -' 1, a subsequence of (O) converges to a harmonic map 0 in 10(M\{a ... , ak}, N). At each of the points a1, the derivatives of (Oj tend

to infinity, and the blow-ups of the sequence of maps approach a harmonic map from R2 to N. The extension theorem (3.47) ensures that it provides a harmonic map from S2 to N. Thus, when a -- 1, 0. approaches 0 except at a finite number of points is represented in n2(N) as the sum of where harmonic spheres bubble off. Finally, various harmonic maps of S2 to N, and not necessarily as a single one.

101

ANOTHER REPORT ON HARMONIC MAPS

417

Also, [Sacks, Uhlenbeck 1] use critical point theory of the Ea to conclude that if the universal cover of N is non-contractible, then there is a non-constant harmonic map 0: S2 --* N.

Note that this does not say that each homotopy class of maps from S2 to N contains a harmonic representative, although no counterexample has been (5.4)

.

produced. However, [Futaki 1] has shown that certain homotopy classes of maps from S2 to a Hirzebruch surface N (of real dimension 4) do not contain any energy minimising harmonic map.

The idea is that rt2(N) = Z p+ Z and that two generators a,/3 of ir2(N) have intersection properties such that a minimising harmonic sphere in a+l would break in two separate spheres, one in a and one in /f (one having bubbled off the other). (5.5) [Struwe 3] has provided an alternative description of the bubbling off process, along the solution of the heat equation (3.40); see also [Chang].

Existence of harmonic maps between surfaces (5.6) Suppose that both (M, g) and (N, h) are compact surfaces, and consider the problem of existence of harmonic maps in the various homotopy classes ' e [M, N].

The only cases not covered by (5.2) are N = the 2-sphere S or N = the real projective plane P. For maps from an orientable surface to S, our knowledge of the existence problem is described in [Report (11.3)-(11.9)]. Similar answers are given in [Eells, Lemaire 2] for maps from a non-orientable surface to S. The homotopy classification of maps from a surface to P is more involved. A given homomorphism 0: irl(M) -+ n1(P) is induced by an infinity of homotopy classes if it is oriented ; otherwise by either one or two classes. Only partial results have been obtained. In particular: (a) If O is induced by one homotopy class, it contains a minimising harmonic map by (3.19) and (3.29). (b) If O is induced by two classes, then both contain harmonic maps whose images are closed geodesics [Adams 2] ; and both contain energy minimising harmonic maps, provided 0 0 0 [lost 10]. It is not known whether the geodesics are energy minimising; this might depend on the metrics g and h. (c) If 0 is induced by an infinity of classes, then at least two contain an energyminimising map [Jost 10]. (5.7)

Consider now the following:

RENDERING PROBLEM.

Let M and N be compact surfaces and Jr e [M, N] a

homotopy class. Do there exist metrics g and h on M and N such that ., contains a harmonic map 0: (M, g) -* (N, h)? (5.8) Combining results of [Lemaire 2, 3], [Eells, Wood 1], [Eells, Lemaire 2] and [Adams 2] shows that the answer to that problem is yes-except in the following four cases, where the answer is no:

(a) M = T = the 2-torus, N = S and Ideg (.*')( = 1; 14

BLM 20

102

418

J. EELLS AND L. LEMAIRE

(b) M = T, N = P and A' is the class obtained from that in (a) by composition with the covering map 7t: S -+ P;

(c) M = P, N = S and ,Y is the non-trivial homotopy class; (d) M = P, N = P and ,Y is the class obtained from that in (c) by composition with it.

Here deg ($") is the common degree of the maps 0 e X. (5.9) Let j,r denote the index of the subgroup c*(rt,(M)) in rz,(N). [Edmonds] has given the following solution to the rendering problem for holomorphic maps. Let M and N be compact orientable surfaces and . a homotopy class with deg (.f) : 0. For any complex structure on N, there is a complex structure on M relative to which X contains a holomorphic map if and only if Ideg (°)I > j,e or c* : tt,(M) -+ 7t,(N) is injective.

Harmonic diffeomorphisms (5.10) [Jost, Schoen]. Let (Mg) and (N, h) be homeomorphic closed surfaces. Any diffeomorphism yr : M -+ N is homotopic to a harmonic diffeomorphism 0 of least energy amongst all diffeomorphisms homotopic to yp.

[Coron, Helein] have shown that 0 is energy minimising amongst all maps in the homotopy class of yi. Also, 0 is unique if genus (M) >, 2. When Riem' < 0, it had previously been established by [Sampson 1] and [Schoen, Yau 2] that the unique harmonic map in the class is a diffeomorphism. (5.11) Previous attempts ([Shibata], [Sealey 1, 4]) to prove (5.10) led to this question. If o e'° n £i(M, N) is a quasi-conformal homeomorphism for which

is holomorphic as a distribution on M, is 0 harmonic? If 0 were smooth that would be true [Report (10.5)], but doubt is expressed by the following result of [lost 11]. Any continuous map yr : (M, g) -+ (N, h) is homotopic to a map o e r ° n 2' (M, N)

is holomorphic. Jost also constructs such a Lipschitz map

for which

0: T -+ S of degree 1, which by (5.8) is certainly not harmonic. However, [Helein 2] has just established that the answer is yes. (5.12)

Let M, N be homeomorphic surfaces of common genus >, 1. Then for any

c > 0, {0 e fi(M, N) : ¢ is a homeomorphism and E(O) < c} is equicontinuous [Lelong-Ferrand]. (5.13) In contrast to (5.10), a construction of [Calabi 3] implies [Eells, Lemaire

3] that there exist smooth metrics g on the 3-dimensional torus T3 such that any harmonic map 0 of (T3,g) to the flat torus (TI, g,) has singularities (that is, points where the Jacobian J. vanishes). Minimal surfaces (5.14) Recall [Report (10.5)] that any map 0: (M, u) -+ (N, h) from a Riemann surface to a Riemannian manifold which is conformal (2.26) and harmonic is minimal in the sense that it is an extremal of the area (that is, volume) functional (2.21) and

is a branched immersion. The local structure of branch points of area minimising conformal harmonic maps has been analysed by [Micallef, White].

103

419

ANOTHER REPORT ON HARMONIC MAPS

When M = S2, any harmonic map is conformal.

(5.15)

(5.16) When genus M ? 1, an extension by [Sacks, Uhlenbeck 1] of classical results of [Douglas 1, 2, 3] and [Morrey 1] asserts that a map 0 which is an extremal of E for all variations of the conformal structure on M and all deformations of 0 is conformal and harmonic. (5.17)

This was used by [Sacks, Uhlenbeck 2] and [Schoen, Yau 3] to prove that

if M is a surface, (N, h) a manifold, and if 0: tr,(M) -* ir,(N) is an injective homomorphism, then there exists a conformal structure p on M and an area minimising conformal harmonic map 0: (M, u) -+ (N, h) such that 0 is contained in the conjugacy class of homomorphisms ¢.. When n2(N) = 0, the homotopy class of 0 is determined by 0. The proof requires two minimising processes. In the class of maps inducing 0, first find harmonic maps for given conformal structures p on M; then minimise the energy amongst conformal structures. Supposing for simplicity that M is oriented, the space of conformal structures is identified with the space of complex structures, which is not compact (see (5.47) and (5.51) below). However, if the structures It of a minimising sequence leave all compact sets, then the length of a closed geodesic of M (with the metric of curvature -4 corresponding to Ic; see (5.46)) tends to zero. Analysis of the

energy shows that its image in N tends to a point-thus contradicting the injectivity of ¢*. This can best be illustrated in the case of surfaces with boundary; (see 12.52). (5.18) In general, a minimal map ¢ obtained by (5.17) is a branched immersion. However, if dim N = 3, a method of [Osserman 2], [Gulliver 1] and [Alt 1, 2] shows that

its Jacobian does not vanish, so that it is an immersion. (5.19) Minimal surfaces in the sphere S3 have been constructed by [Lawson 4], who proved that for any compact surface M except the projective plane, there exists a minimal immersion of M in S3, for suitable p on M. If M is orientable, there exists a minimal embedding; and if the genus of M is not prime the embedding is not unique. The construction is made by successively reflecting geodesic polygonal minimal surfaces in S3.

On the other hand, there is no non-constant harmonic map of P to S3 [Lawson 4], [Eells, Lemaire 2]. (5.20) Lawson has also shown that a compact embedded minimal surface in S3 separates it into two diffeomorphic components. In all of his examples, these components have equal volumes.

However, [Karcher, Pinkall, Sterling] have produced new minimal surfaces in S3 separating it in two components of different volumes. Their construction goes by (a) tessellating S3 into cells having the symmetry of a Platonic solid in P83; (b)

dividing each cell by planes of symmetry to obtain a fundamental

tetrahedron ; (c) finding within such a tetrahedron a minimal surface with boundary which intersects orthogonally all plane faces along geodesics; (d) reflecting them across those planar faces not contained in the faces of the

cell; (e)

reflecting across the faces of the cells. 14-2

104

420

J. EELLS AND L. LEMAIRE

(5.21) [Smyth 1] gave such a construction in l3, and proved that every flat torus T3 contains minimally embedded surfaces of arbitrarily high genus. [Meeks 1] makes a detailed study of such embeddings. See also [Micallef 2].

[Pitts, Rubinstein 2, 3] have constructed infinite families of embedded minimal

surfaces in S3; and in other 3-manifolds endowed with Thurston's geometric structures. A basic tool is a minimax procedure [Pitts 1]. (5.22) [Bryant 2]. Any compact Riemann surface M can be conformally harmonically immersed in S'.

Holomorphic curves (5.23) As a special case of his extensive study of holomorphic curves in almost complex manifolds, [Gromov] has produced families of minimal spheres in (P,(C), h) as follows. On P2(C), denote by wo the Kahler form of the usual complex structure Jo and Fubini-Study metric ho. Let h be a Riemannian metric and Jh the almost complex structure for which coo is the Kahler form of (P2(C), h, Jh). Then any two distinct points y, y, e PZ(C) lie on the image of a holomorphic curve 0: S2 -+ (P2(C), JR), homologous to P1(C) c P2(C). By (4.3) this map is conformal and harmonic. More generally, for d >, 2, any d(d+3)/2 points in general position lie in the image of a holomorphic map 0: M --+ (PZ(C), Jh), where M is a surface of genus (d2 - 3d+ 2)/2 and 0 is homologous to dP,(C) c P2(C).

(5.24) Suppose that there is a smooth family (h,)o,,,, of metrics on P2(C) such that ho is the Fubini-Study metric, and that for every t the h,-harmonic 2-form co, generating H2(P2(C)) does not vanish. Then for h = h, (indeed, for any h) there exists a unique almost complex structure

J on PZ(C) for which co,,(X, JY) = I mh12 h(X, Y), so that co,, is the Kahler form of the conformal metric h' = I co,,I2 h. By a result of Moser, the existence of the family (h) implies that there is a diffeomorphism of P2(C) to itself carrying co,, to coo.

Transporting the almost complex structure J by this diffeomorphism and applying (5.23), we see that for such admissible metrics h, there exist large families of minimal maps 0: M -+ (P2(C), h'). That applies in particular if h is close to ho. (5.25) [Gromov] also proves the following compactness theorem for holomorphic curves-in a delicate sense. Let M be a compact oriented surface and (JM)j a sequence of complex structures on M respecting the orientation; let (N, h) be a compact oriented even-dimensional Riemannian manifold, and (J'),,, a uniformly Ck'"-bounded sequence (k 2) of almost complex structures on N with respect to which h is Hermitian ; and let 0,: (M, JM) -+ (N, h, Jl) be a sequence of C" holomorphic maps with V(h) uniformly bounded.

With these hypotheses, we cannot expect subconvergence of the sequence of maps and Riemann structures. Indeed, maps of spheres might bubble off at a finite number of points as in (5.3), and geodesics in M might get pinched to points as in (5.17), if JM leaves all compact sets of the space of complex structures. By suitably interpreting and controlling these two processes, [Gromov] was able to conclude that the sequence (0) subconverges (in an appropriate C' sense) to a holomorphic map of a compact Riemann surface (M, J) to (N, h, JN'), where M is obtained from M by pinching a finite

105

ANOTHER REPORT ON HARMONIC MAPS

421

number of Jordan curves to a point, f is the C'-limit of a subsequence of the Jr", and JN of the JN. The surface (M, J) can be interpreted as a point in a compactification of the space of complex structures. Surfaces of parallel mean curvature (5.26) For each p, (5.19), (5.20) and (5.22) provide embedded minimal surfaces of genus p in S3. Composing them with the canonical embedding S3 -+ 118' yields surfaces of parallel mean curvature of every genus. Their Gauss maps y : M --> Gz(R') (see (2.35)) decompose as y = (y+, y_), using the Riemannian product structure

G2 (R') = S+ x S_, where S+ and S_ are 2-spheres. By a result of Blaschke (see [Chern, Spanier]), the degree of y+ and y_ is I -p. Thus [Eel]s, Lemaire 2], for every oriented surface of genus p, the homotopy class of maps of degree 1- p of Al to S can be rendered

harmonic by a Gauss map yt. (5.27) Analogously, the non-trivial homotopy class of maps from the Klein bottle K to S is represented harmonically via the Gauss map of Delaunay's nodoid in I83 [Eells, Lemaire 2], using the identification GZ(I83) = S.

(5.28) Given an oriented surface M of constant mean curvature in I83, its Gauss map is harmonic from M to S2. Conversely, for any harmonic map y from a simply connected surface to S2 and for any positive number H, [Kenmotsu] has obtained an explicit representation of a branched immersion 0: M -+ R3 with constant mean curvature H such that y, = y. The representation of 0 is obtained by means of the following Weierstrass type formula. Let y' denote the map from M\y 1(0, 0, 1) to R1 obtained by composing y with the stereographic projection and set _2y, i(I+Y,2) A

H(l+Iy'I2)2'H(1+IY 12)2'H(1+Iy'2)2

Then 0 is given by Z

O(z) = 2 Re J A yz + c. (5.29) If M is not simply connected, the above formula does not often provide a surface in I83, because of difficulty with the periods of the integrals. In fact, we have two classical results : (a)

Any isometric immersion 0: S2 - R' with constant mean curvature is an

isometric embedding onto a Euclidean sphere [Hopf]. (b) Any compact isometrically embedded hypersurface 0: (M^-', g) --+ W of constant mean curvature is a Euclidean (n -1)-sphere [Alexandrov].

(5.30)

However, [Wente 1, 2] has shown that certain tori (T2, g) can be

isometrically immersed in R' with constant mean curvature.

The idea of his proof is to solve an appropriate boundary problem for the sinh-Gordon equation on a rectangle, in such a way that the solutions on two such rectangles can be pieced together to give a surface of constant mean curvature on their

106

422

J. EELLS AND L. LEMAIRE

union. All this depends on a parameter which can be adjusted so that after adding a certain number of rectangles, the surface closes to form a torus. (5.31)

[Abresch] gave an explicit representation of some of these surfaces, in terms

of elliptic functions-specifically, those for which the curvature lines for the smaller principal curvature are planar. That property leads to a separation of variables in the sinh-Gordon equation. The Gauss maps of Wente's surfaces are harmonic non-holomorphic maps from T to S, necessarily of degree 0. (5.32) Recently, [Kapouleas] has shown that for any p >, 3, there are infinitely many orientable surfaces of genus p immersed in R3 with constant mean curvature. The main idea of his proof is as follows. Recall [Report (11.7)] that the unduloid

and nodoid of Delaunay are complete surfaces of revolution and constant mean curvature in R3. They can be parametrised by a real number a, so that a < 0 for the unduloid and a > 0 for the nodoid. When a tends to 0, the regions of positive curvature tend to spheres and those of negative curvature-suitably blown up-to catenoids (the surface corresponding to a = 0 is a string of spheres each tangent to the next one). Starting from a configuration of spheres linked together by pieces of Delaunay surfaces, Kapouleas deforms it to a nearby surface of constant mean curvature. The existence of a solution to the deformation equation follows from an implicit function theorem. To solve the linearised equation, a balancing condition must be imposed on

the configuration: namely, for each sphere the sum of the vectors pointing in the direction of the various Delaunay pieces, multiplied by functions of the associated of, must be 0. For instance, a constant mean curvature surface of genus 3 could look roughly like 4 spheres at the vertices and centre of a triangle, linked together by 3 nodoids and 3 unduloids. The case p = 2 escapes that construction, because the balancing condition cannot be satisfied. (5.33)

Various results on minimal or constant mean curvature immersions are

established by constructing a holomorphic object (for example, a quadratic differential) and using its properties. [Eschenburg, Tribuzy] have observed that holomorphicity was only used through the behaviour near the zeros, and that these results could therefore be generalised. For example, they prove the following extensions of (5.15) and (5.29a).

Let 0: S2 - (N, h) be a map such that Iz(g5)l <,f 1(c*h)2.°I for some function feWOO(S2). Then 0 is conformal and harmonic. Let 0: S2 -+ I1 be an isometric immersion such that Idlr(cb)I I <_ JrI2-4K), where K .

is the Gaussian curvature and fEWw(S2). Then 0 is an embedding onto a Euclidean sphere. Minimal embeddings

(5.34)

In (5.19) and (5.20), minimal surfaces are obtained in a rather constructive

way, from which we see that they are embedded. We cannot usually draw such a conclusion from the existence proofs by minimising sequences in (5.3) and (5.17).

107

ANOTHER REPORT ON HARMONIC MAPS

423

However, we shall see that for area minimising surfaces, self intersections tend to be as small as their homotopy classes allow. Roughly speaking, unnecessary folds can be cut off by surgery, yielding a new surface of smaller area. (5.35) The following sphere theorem was obtained by [Meeks, Yau 1]. Let (N, h) be a compact 3-dimensional manifold, such that ir2(N) 0. (a) There exists a family 01, ... , g,e : S2 -. N of conformal harmonic maps such that {01, ... , O k} generates 7r2(N) as a iz,(N)-module, 0, minimises the area amongst all homotopically non-trivial maps; and for r = 2, ... , k, 0, minimises the area in the complement of the n1(N)-module generated by ...,0,_1}.

For each family

(b)

y/,} satisfying those properties, V. is either an

embedding or a double covering of an embedded projective plane. (c) For each r and s, yr8(S2) and 0,(S2) are either equal or disjoint. The existence statement (a) is a refinement of (5.3). The proof of (b) involves an

approximation argument to reduce the problem to the real analytic case, the tower construction, and a surgery argument to suppress unnecessary folds. (5.36)

This result provides a new proof of the sphere theorem in topology,

yielding a representation of generators of rc2(N) by smoothly embedded spheres. (5.37)

The following embedding theorem (with several variations) is due to

[Freedman, Hass, Scott], related to a theorem of [Meeks, Simon, Yau]. Let (N, h) be a 3-dimensional compact manifold. Assume that N is P2-irreducible (that is, there is no 2-sided embedded real projective plane in N, and every embedded 2-sphere bounds a 3-

cell). Let M be a compact surface other than S2 or P2. Let 0: M -- N be an area minimising map, homotopic to an embedding yr: M --+ N with trivial normal bundle and such that 0* : n1(M) -+ 7t1(N) is injective. Then either (1) (2)

0 is an embedding; or 0 double covers a one-sided surface F embedded in N, and yi(M) is the boundary

of a region in N which is a twisted I-bundle over a surface isotopic to F. Examples show that the second case can actually occur. (5.38)

If 0, V : M1 --* N and 02, yr2 : M2 --* N are as above, and if yr,(M) is disjoint

from yr2(M), then q31(M) and 9S2(M) are either disjoint or identical.

The main ingredients in the proof are geometric in nature. They involve in particular a study of the regions bounded by pieces of ¢(M), or by 01(M) and 02(M) (5.39)

[Freedman, Hass, Scott] then proceed to study the homotopy classes in

which every map has a self-intersection. Using a count of components in an appropriate covering, they define a measure of self-intersection D(O) and a measure

of intersection D(q1, 0,) for two surfaces. They show that if 0, and 02 are area minimising and respectively homotopic to yr, and yr2, then

D(pi) < D(vl) and D(01, 0) < D(yi1, W2). However, except when 0, or 02 is a map of a torus or Klein bottle, the area minimising maps do not necessarily minimise the number of double curves in their self-intersection or intersection.

108

424

J. EELLS AND L. LEMAIRE

Area non-minimising embeddings (5.40)

When n2(N) = 0, each map of S2 to (N, h) is homotopic to a constant, so

that non-trivial area minimising maps cannot exist. Nonetheless, there are many interesting embeddings which are extremals of the area functional. Proof of their existence has required use of different techniques, in particular mountain pass lemmas and geometric measure theory. (5.41) [Simon, Smith]. For every metric h on S3 there is a minimally embedded 2-sphere in (S3, h). [Meeks, Simon, Yau]. If Scales',') > 0, then any two embedded compact minimal

surfaces of the same genus are isotopic. (Special cases with restrictions on the Ricci curvature were first obtained by [Lawson 5] and [Meeks 5].) (5.42) [Pitts 1].

There is a minimally embedded closed surface in any compact

manifold (N3, h). More specifically, the Heegard genus of N3, H-genus (N3), is the least integer p for

which there is a closed smooth embedded surface M of genus p such that N\M has exactly two components, which are handlebodies. Here the surface M could be nonorientable, in which case p is the number of its crosscaps. Then [Pitts, Rubinstein 1, 2] : if (N, h) is a compact oriented 3-dimensional manifold, there is a minimal embedding

0: M -+ N of a surface M such that genus M < H-genus N. Moreover V-index (0) < I < V-index (0) + V-nullity (0).

The case H-genus = 0 is (5.41). If (N, h) = (P3(OB), h), then H-genus N = 1; and there is an embedded surface (5.43) EXAMPLES.

which is either S2, T2 or P2(O1). However, if P3(R) carries its canonical metric, [Frankel] has shown that there is no minimal embedding of S2 in P3(R).

His proof rests on the following fact. Let (N, h) be an n-manifold with Ricci('," > 0, and Ml, M2 be minimally immersed (n-1)-manifolds, one of which is

compact. Then Ml n M2 : 0. (5.44)

[Pitts 1] extended (5.42) to higher dimensions by showing that every

compact manifold (N, h) with n = dim N < 7 contains a compact minimally embedded (n - ] )-manifold (the case n = 7 is due to Schoen and Simon). (5.45) Finally, let (N, h) be compact with Ricci'"' > 0. Then [Choi, Schoen] the space .p of compact embedded minimal surfaces of genus p in (N, h) is C'"-compact

(for any k >, 2). There is a constant depending only on (N, h) and p such that max JVddI s c for any 0 e Yp. Furthermore, if (N, h) is real analytic, then Sop is a finite dimensional real analytic variety. Teichmiiller space (5.46) Let M be a compact oriented surface of genus M = p > 2. (The cases p = Oandp = 1 are simpler, but should be treated separately.) The space .11(p) oforiented

complex structures on M is canonically identified with the space of conformal structures, and with the space of Riemannian metrics of constant curvature -4.

109

ANOTHER REPORT ON HARMONIC MAPS

425

Given its C°°-topology, .df(p) is a contractible complex manifold modelled on a separable Frechet space. The diffeomorphism group -9(M) of oriented diffeomorphisms acts continuously, effectively, and properly on #(p):

.i(p)x_q(M)->,#(p)

by

Let _q°(M) be the closed normal subgroup of diffeomorphisms homotopic to the

identity. It turns out through the following construction that -9°(M) is the arc component of the identity in 2 (M). (5.47) [Earle, Eells 1, 2]. If n: , #(p) -, (p) = 4(p)/_90(M) is the quotient map with quotient topology on .l(p), then it is a locally trivial principal 2°(M)-bundle. The space 9-(p) is the Teichmuller space of M. The quotient 9?(p) = ..#(p)/_9(M) is the Riemann moduli space of M. It represents the space of non-equivalent oriented complex structures on M. The space .gy(p) is the universal covering space of the (singular) space R(p), whose fundamental group is the Teichmuller modular group .9(M)/_9°(M):

.l (p)

R(p) (5.48) Fix a complex structure JE.,K(p), and let sd° be the complex vector space of smooth differential forms on the Riemann surface (M, J) of type (0, q) with values

in the holomorphic tangent bundle T'(M,J). Then .41 = kera, = T,..ll(p), the tangent vector space of.,W(p) at J; and sd° is the vector space of smooth vector fields

on M. Denote by 0, the sheaf of germs of (1, 0)-fields of (M, J). Then, using the differential dir(J), the quotient

sill.41 = H1(M, 0,,) = T .,j) °T(p) is identified with the Kodaira-Spencer space of infinitesimal deformations of the Riemann surface (M, J). Furthermore, T,,(,,) J-(p) = . (M, J), the space of holomorphic quadratic differentials on (M, J) ; in particular, dime 9-(p) = 3p - 3. Thus we get a direct sum decomposition T, ..K (p) = Ker do ® PJ(M, J). Let g and h be metrics of constant curvature -4 on M; thus g,he.A'(p). In the homotopy class of the identity map M --+ M, there is a unique harmonic map (M, g) - (M, h), which is a diffeomorphism (5.10). Moreover, it depends smoothly on g and h (2.18). This permits the following construction [Earle, Eells (5.49)

2].

(5.50) Fix g. Since a harmonic map composed with an isometry is harmonic, we have 0 o 0, 6, = c, ,, for all 0 e -9°(M). Therefore, the map a: 9-(p) --- . &(p) given by

110

426

J. FELLS AND L. LEMAIRE

h is well defined, independently of the choice of h in rt-'(rr(h)). a(n(h)) varies smoothly with h, and defines a section of it:

a(rc(h))

a M(P)')7t-'T (P) Thus we have a smooth global trivialisation of the fibre bundle -&(P) =

(P) X 2o(M)-

Application of (5.51) below shows that the group 24(M) is contractible. (5.51) Again holding g fixed, define the map Q : (p) - M(M, g) by Q(p) = (Op ,, where a(p) = h. As an analogue of Teichmuller's parametrisation,

we have the following result: Q is a diffeomorphism of (p) onto -V(M,g). [Sampson 1] showed that Q is continuous and injective. That it is surective was proved by [Hitchin 3], using his self-duality equations; and by [M. Wolf], using the properness of the function E:.9 (p) - ll defined by E(,u) = E(Og,,,). [Schoen, Yau 3] proved that E is proper with the roles of g and h interchanged.

(5.52) Now take any q 0 0 in R(M,g), and form the ray (tri),,0. Then Q-101) = p(t) is a path in .T(p) which leaves every compact set. Thurston has compactified !'(p) by adjoining to it the topological (6p - 7)-sphere of projective measured foliations ([Fathi, Laudenbach, Poenaru] give full details). [M. Wolf] has characterised that construction as follows. Thurston's compactfication of (p) is obtained by adjoining all points corresponding under Q-' to the points at infinity of the rays of .94(M, g). Wolf's proof is based on a detailed study of the asymptotics (as t -- oo) of the Beltrami differentials ¢{t)/q=(t) of the harmonic maps 0(t) determined by p(t) = Q-'(trl). (5.53)

The Petersson inner product on PJ(M, J) is defined by

J *j1 dx, M

where (M, J) carries its compatible metric of curvature - 4 and dx is the corresponding volume element.

Identifying a(M, J) with the tangent space of .l(p) at rr(J), this furnishes a Hermitian metric on the complex manifold -q-(p), called the Petersson-Well metric, which in fact is Kahler [Ahlfors]. [Schumacher] has used an integral formula of [Wolpert 2] to show that it has strongly negative curvature tensor in the sense of (4. 10). Treating the Riemann moduli space 9P(p) as a Kahler orbifold, he concludes that certain harmonic self-maps of A(p) are holomorphic. (5.54) With E defined in (5.51), the Hessian d2E(u°) at the unique minimum p0 is nondegenerate, and its Hermitian extension coincides with the Petersson-Well metric at

µ0. That fact is due to [Fischer, Tromba], in a form with the roles of g and h interchanged.

(5.55) A neighbourhood of u0 in 9-(p) can be parametrised by a neighbourhood of 0 e M(M, g). Relative to a unitary base, for t e C31-3 the Riemann curvature tensor

111

ANOTHER REPORT ON HARMONIC MAPS

R of the Petersson-Weil metric on

427

(p) satisfies [M. Wolf]

34E(p(t)) ata aT aty ai' I,_o

= -4RakC

Note that only variations of the target are used.

The Petersson-Weil metric is not complete [Wolpert 1]; however, its geometry is rather satisfactory. For instance: (5.56)

The function E:

(p)

R is

(a) proper and strictly plurisubharmonic (4.32), which gives a proof of the

theorem of Bers and Ehrenpreis that (b)

(p) is a Stein manifold ;

strictly convex along geodesics (that is, if c: 11-..T(p) is a non-trivial

geodesic, then d2E(c(t))/dt2 > 0). That easily implies Kerchoff's theorem that the action on (p) of any finite subgroup r of the group _9(M)/_9o(M) (= the outer automorphism group of ir,(M)) has a fixed point p0. Indeed, y, is the unique minimum

of the function E averaged over F. These results are due to [Wolpert 3] with E replaced by a geodesic length function ; and then to [Tromba 1, 2] for the energy function itself. (5.57)

Let M and N be two Riemann surfaces of the same genus p > 2.

Teichmuller's original approach to the moduli problem was to prove the existence of a Teichmuller map 0: a quasiconformal homeomorphism from M to N minimising the maximum of the dilatation. The map satisfies Beltrami's equation, whose coefficients involve the holomorphic quadratic differential appearing in (5.48), and can be interpreted as a harmonic map with respect to degenerate metrics on M and N, associated to the relevant holomorphic quadratic differentials (see [Report (11.15) and (11.16)]). [Leite] gave a characterisation of Teichmuller maps in those terms, and proved that if M and N are two Riemann surfaces and N carries a degenerate metric associated to a holomorphic quadratic differential, then every non-trivial homotopy class contains a Holder continuous harmonic map. 6. Second variation Stability of the identity map

(6.1) A harmonic map 0: (M, g) - (N, h) is said to be E-stable if and only if Eindex (0) = 0, or equivalently if its Hessian (2.14) is > 0. A local minimum is stable; however, a stable map is not necessarily a local minimum, the condition being only a second order restriction. Similarly (2.23), a minimal immersion 0 is V-stable if V-index (0) = 0, (6.2) We now consider the index (= E-index) and stability properties of the identity map Id,: (N, h) -> (N, h)-a local version of the problem described in (2.5).

(6.3)

Let (N, h) be oriented and compact; and c (resp. i) its Lie algebra of

infinitesimal conformal vector fields (resp. the subalgebra of infinitesimal isometries).

Then [R. T. Smith 2] for n > 3 index (Id,) > dim (c/i). For instance, index (Id,.) = n + 1 if n > 3.

112

428

J. EELLS AND L. LEMAIRE

(6.4)

If (N, h) -+ (N, h) is a finite Riemannian covering, then index (IdN),< index (IdN);

for if v is a variation on (N, h) with H,d,(v, v) < 0, then its lift v has H,d',(v, vv") < 0. If 0: (M, g) -* (N, h) is an isometric immersion, then index (0) 3 index (IdM). (6.5) Suppose that (N, h) is a compact oriented Einstein manifold ; thus Ricci'N "' = c-h with c = Scall'- /n, a constant for n >- 3. Then [R. T. Smith 2]

index (IdN) = number of eigenvalues d of A such that 0 < A < 2c; nullity (IdN) = dimension isometry group of (N, h) +multiplicity of the eigenvalue 2c,

where A is the Laplacian on functions on N. (6.6)

Starting from that, [Ohnita 1] and [Howard, Wei] obtain the following

characterisation ; the equivalence (b) _ (d) is also due to [Pluzhnikov 4], and (b) = (c) to [Tyrin 2]. Let (N, h) be a compact irreducible homogeneous space. (Thus N = G/H and the isotropy representation of H on TH(N) is irreducible.) The following properties are equivalent : /n, where A, is the smallest eigenvalue of A. (a) 11 < 2 (b) IdN is unstable. (c) Every stable harmonic map .: (M, g) -> (N, h) from a compact manifold is

constant. (d) Every stable harmonic map yr: (N, h) -- (P, k) is constant. The proof is based on a suitable standard minimal immersion of (N, h) in a Euclidean sphere, using eigenfunctions of A. In that regard, see also [Xin 2] and [Pan, Shen]. (6.7) If (N, h) satisfies the above conditions, then rt1(N) = 0 = ir2(N), as we see by applying [Report (5.3) and (10.9)]. That implies [Urakawa 1] that for any finite group F (0 e) of isometries acting freely on a compact irreducible homogeneous space (N, h), the quotient Riemannian manifold (N/I', h) has stable identity map. (6.8) ExAMPLEs. [Ohnita 1]* has classified the compact irreducible symmetric spaces with unstable identity map. They are

(i) the simply connected simple Lie groups (A")",1; B2 = C2; (C")",3. (ii) (iii) (iv) (v) (vi)

SU (2n)/Sp (n) (n >, 3).

spheres S' (n >, 3). quaternionic Grassmannians Sp (p+q)/Sp (p) x Sp (q) (p >, q > 1). Ee/F4. the Cayley plane F4/Spin (9).

Partial lists had been given by [R. T. Smith 2] and [Tyrin 2]. (6.9)

That any non-constant harmonic map yr: S" --> (N, h) has index (w) > 0 for

n > 3 is due to [Xin 1]; in fact [Eells, Lemaire 4]: index (V/) > k+1 where k is the maximal rank of yi. [Leung 1] has shown that any non-constant harmonic map

0: (M, g) -> S" (n > 3) of a compact manifold has index (0) > 0. The totally geodesic embedding 0: S m --> S' for n > m > 3 has index (0) = n + 1. *and H. Urakawa, Comp. Math. 59 (1986) 57-71.

113

ANOTHER REPORT ON HARMONIC MAPS

(6.10) EXAMPLE [R. T. Smith 2], [Urakawa

1].

0, then index (Id,,,) = 0

and

429

If (N, h) is compact and

nullity (IdN) < n.

(6.1 1) EXAMPLE [Howard].

For all n > 3 there is a number 8(n) with 1 /4 < b(n) < I such that if (N, h) is simply connected with b(n) < Riem(N ") S 1, then for every compact (M, g) every stable harmonic map 0: (M, g) - (N, h) is constant. With any Hermitian complex structure on S2D+1

(6.12) EXAMPLE.

X

S2v+1 (p

3 1,

q >, 1), the identity map is harmonic, holomorphic, and index (Id) > 0. Spectral analysis (6.13)

The inverse problem of spectral theory consists in recovering properties of

an operator from the knowledge of its spectrum. For the E-Jacobi operator that programme has been undertaken by [Urakawa 2], using calculations of Gilkey. Let 0: (M, g) - (N, h) be a harmonic map with M compact; and let

Y. exp (- tpj) - (4nt)mi2

ak(Jo)

J-1

tk/

be the asymptotic expansion of JJ = J,', .(p) denoting its spectrum. Then ao(J4) = n Vol (M, g)

a,(J,) a2(J4)

Ricci",'`')] dx

6 IM

360

[5(Scal(M_ 2RicciM)112+ 211 RiemM)112 dx / IM

+360

f",[-30jj0*Riem("- "'112+60 Scal(M'g) Traceg(q5*

180110* Riem(N. " )112] dx.

These lead to various characterisations, such as the following. A harmonic map ¢: S"--> S' with the same Jacobi spectrum as an isometry must be an isometry. (6.14)

A harmonic map 0: (M, g)

(N, h) with

the same spectrum as a geodesic

yi : S' -+ (N, h) must be a geodesic. Let Scal(M,') be constant, and suppose that (N, h) is a space form of non-zero constant curvature. Then any harmonic map 0: (M, g) -* (N, h) having the same spectrum as a minimal isometric immersion must be a minimal isometric immersion.

Any harmonic map 0: S2 -+ P"(C) with the same spectrum as a holomorphic isometric immersion yr coincides with V/ up to an isometry.

On the other hand, if 0, 0' are two harmonic maps of compact manifolds (M, g), (M', g') to flat tori T, T', then spectrum (JJ) = spectrum (J0.) if and only if dim (T) = dim (T') and spectrum (A) = spectrum (A'), where A and A' are the Laplacians on M and M'. Thus in general the spectrum of J¢ does not characterise the domain (M, g) or the map 0.

114

430

J. EELLS AND L. LEMAIRE

(6.15)

For any map 0 : (M, g) - (N, h) of a compact manifold, letl R g'= sup sup { E : v e TT(X)(N), I vl = 1 } . 2EM

t-1

JJJ

Note that, for B > 0, RO < 2B sup {e,(x) : x e M} if Riem(",'" < B.

Let ZM(t) = E°°oe tz+ be the trace of the heat kernel of the Laplacian acting on functions on M. Using a zeta function estimate of [Hess, Schrader, Uhlenbrock], [Urakawa 1] has shown that for any harmonic map 0: (M, g) - (N, h), 0 < t < oo}. index (¢)+nullity (0) < (6.16) APPLICATION.

a > 0 and 0: (M, g) - (N, h) is harmonic,

If Ricci("

then

index (¢)+nullity(q) ` n 1 + n

(1 +4a2) if m = 2 and a = R"/a

+a {1+(m-l)!mm-la(l+cxr-1) if m>,

3anda=m-1R0/a. m

These estimates are in the spirit of Morse and Schoenberg for M = S'; however, sharper estimates exist in that case. There are analogous results for the volume functional and the V-index and nullity. (6.17) Let 0: (M, g) -- (N, h) be a Riemannian submersion with totally geodesic are the projections on the horizontal fibres. Then [. J', Y''Jo] = 0, where . ' and and vertical distributions. Therefore [Urakawa 1], t'2(c-'T(N)) has a complete orthonormal basis of simultaneous eigensections with respect to J', .WJ' and Also, index (0) >, index (IdN), nullity (0) > nullity (IdN).

J.

Morse index theorem (6.18) The following version of the Morse index theorem has been established by [Ishihara 1]: Suppose that (Mg) is compact with boundary 8M, and let v be the outward unit normal field. Then the Hessian of a harmonic map is

<J(v), w> dx+

HH(v, W) = TM

fam

Let '0(q-1T(N)) = {ve '(c-'T(N)): v = 0 on 8M}. The Jacobi fields along 0 are ve(6o(0-1T(N)) such that J0(v) = 0; the nullity of 0 is the dimension of now those Ker Jo c'o(O-'T(N)), and the index of 0 the dimension of the maximal subspaces of 'o(q-'T(N)) on which HH is negative definite. Thus index (0) is the sum of the multiplicities of the negative eigenvalues 2 of the problem JJ(v) = 1v on M, v e'o(0-'T(N)).

115

431

ANOTHER REPORT ON HARMONIC MAPS

Let i : M x I M be a smooth map with i, = identity, and such that M, M, for t > s, where M, = i,(M). Say that i is of e-type (e > 0) if Vo1(M, i,* g) < e for all t near 1. Say that 8M, is a conjugate boundary of 0 if nullity (c 1m) > 0. (6.19) If 0: (M, g) -+ (N, h) is a harmonic map, there exists e > 0 such that if (0, has e-type, then the number of conjugate boundaries is finite and

index (0) _

nullity (0 IM). o
(6.20)

An analogous theorem for the volume functional has been given by

[Simons] and [Smale].

Harmonic variations

(6.21) A variation v of a harmonic map q!: (M, g) - (N, h) is harmonic if = exp (tv) is harmonic for all t e R. Such variations were introduced by [Toth 1, 2] and used in his study of local rigidity of maps. For instance: If M is compact and 0: (M, g) -+ RN(c) is a harmonic map into a real space form with c -A 0, then v is a harmonic variation if and only if v is a Jacobi field with IvI constant, and <do, Vv> = 0. (6.22) If v is a Jacobi field along a harmonic map 0: (M, g) -+ S, then either (1) 0Z is harmonic for all teR and IvI is constant; or (2) IvI is a non-zero constant and 0, is harmonic if and only if t = kn/Ivl for some

keZ; or (3)

Ivi is non-constant and there exists at most one t > 0 with

harmonic.

Curvature of 3-manifolds (6.23) Here is an important application of harmonic maps to the geometry of Riemannian manifolds, due to [Schoen, Yau 3]. Let N be a compact orientable 3-manifold; assume that n1(N) contains either a

finitely generated non-cyclic abelian subgroup, or a subgroup isomorphic to the fundamental group of a compact surface of genus > 2. Then N does not admit any metric

h such that Scal" > 0; and any metric with Scal' >, 0 must be flat. (6.24) The pattern of the proof is that discussed in (4.16). For instance, consider the second hypothesis on n1(N). By (5.17) and (5.18), there is a surface M and an area minimising immersion 0: M-+N, such that 0* is the injection of the given subgroup of n1(N). Using an orthonormal frame field (e1, e2, e3) along 0(M) with e31 O(M), the condition H'(e3,e3) > 0 (since 0 is area minimising) becomes 2

JM

(R313+82323+ E IVdc(ei,ej)12 dx

0.

Using Gauss's formula, the Gauss-Bonnet theorem, and the minimality condition, we can write that in the form (Scal"+ jIlVd0(e, e)12) dx < 0,

showing that Scal" > 0 is not possible-the first part of the statement.

116

432

J. EELLS AND L. LEMAIRE

(6.25) An extension of this method to the non-compact case (using a minimal surface constructed by other means) led to the proof of the positive mass conjecture in general relativity [Schoen, Yau 6]. (6.26) The following rigidity theorem corresponding to (6.23) was obtained by [Fischer-Colbrie, Schoen]. Let (N, h) be oriented and have Scal" >, 0; and suppose that M is a compact oriented V-stable minimal surface in N. Then M is a Euclidean sphere or a totally geodesic flat torus. (6.27) The next structure theorem (and its extension to the case where N has a convex boundary) was established by [Meeks, Simon, Yau], using their existence theorem (5.37), the second variation formula (2.23) and various geometric arguments. Let (N, h) be a compact orientable 3-manifold with Ricci("> 0. Then either (1) N is covered by an irreducible homotopy sphere; or

(2) N is dfeomorphic to a handlebody; or (3) N is covered by a Riemannian product S2 x Sl, where S2 carries a metric of non-negative curvature; or

(4) N is flat, and thus N = T3/1

.

Curvature operator in higher dimensions

(6.28)

The curvature operator 9t: A2T(N) --+ A2T(N) is the self-adjoint operator

defined at each point by <98(X A Y), Z A T> = .

We still denote by 9P its complex linear extension to A2Tc(N). (6.29) Let ¢: M (N, h) be a harmonic map from the compact Riemann surface M to the Riemannian manifold (N, h). Let w be a section of O Tc(N). Then the Hessian of 0, extended as a Hermitian form on W(c-1Tc(N)), can be expressed as

[Moore 31: HH(w, w) =

4JM

IDalar wF2 -

(gP I

A w),

LO A w) } dx.

(6.30) A complex linear subspace V c-- Tac, N is called totally isotropic if <X, X> = 0 for all X e V. Say that A is positive on totally isotropic 2-planes (written gP 1T, > 0) if > 0 for every basis X, Y of every totally isotropic 2-plane in Tc(N), for all y e N. Note that Riem" 'I > 0 if and only if the curvature operator A is positive on real 2-planes.

(6.31) [Micallef, Moore]. Let 0: S2-*(N,h) be a non-constant harmonic map; and suppose (N, h) satisfies A ITI > 0. Then index (0) > (n - 3)/2.

This is established by applying the Birkhoff-Grothendieck theorem (4.9) to ® L, and use the Hermitian product H on 0-'TC(N). We have 0-'T'(N) Tc(N) to conclude that cl(Li_i+1) _ -cl(L,). Then (4.9) implies the existence of a number of holomorphic sections, which together with 80/8z span totally isotropic 2-planes. Then (6.31) follows from (6.29).

117

433

ANOTHER REPORT ON HARMONIC MAPS

(6.32)

[Micallef, Moore] refine (5.3) as follows. Let (N, h) be a compact

manifold with lrk(N) 0 0 for some k >, 2. Then there exists a non-constant harmonic map 0: S 2

N of index
This is obtained by using the perturbed energy E.,1 0) = EG(O) + f. dx. M

Here N is isometrically embedded in some Euclidean space Q8° and f maps M to R. (6.33) Combining (6.31) and (6.32), [Micallef, Moore] conclude that if (N,h) is a compact simply-connected Riemannian manifold of dimension n > 4 such that

11.r, > 0, then ;c;(N) = 0 for i < n. Using the affirmative solution of Poincare's conjecture in these dimensions, it follows that N is homeornorphic to S". (6.34) That remarkable application includes a generalisation of the sphere theorem of Berger and Klingenberg, which asserts that if M is compact, simplyconnected and if 1/4 < Riem",') < 1, then M is homeomorphic to a sphere. Indeed, [Micallef, Moore] use an inequality of Berger to conclude that if there is a positive function B: N-+ R such that at each point xeM, B(x)/4 < Riem'",")(x) < B(x), then '

ITI > 0.

(6.35) Theorem (6.33) was obtained independently by Micallef and Moore--the former using the index of minimal maps instead of harmonic maps; and thereby

getting a slightly different curvature condition.

In fact, if 0: M - (N, h) is a minimal map of a surface, then using (2.24) and (6.29) gives the following formula of Micallef:

H.'(v,v)=4 JM {I(Val,v)'

('0 Z-RAvl, Oz / Oz

vI-Vaiazv)TI2}dx. JJJ

(6.36) For any conformal harmonic map 0: M -+ (N, h) of a Riemann surface, we have E-index (0) < V-index (0) with equality in case genus (M) = 0 (Micallef ). In particular, a minimal immersion ¢: M -+ (N, h) is E-stable if it is V-stable.

(6.37)

Refining their analysis, [Micallef, Moore] show that if (N, h) is compact

and either even-dimensional with 9P IT, 3 0, or odd-dimensional with sectional curvature 0 on these 2 -planes with basis {X, Y} such that <X, X> = 0 = <X, Y>, then any stable

harmonic map 0: S2 - (N, h) is real isotropic (in the sense of (7.7) below). As an application, they conclude that if (N, h) is compact, odd dimensional and such that B(x)/4 < Riem","'(x) < B(x) for all x, then rz2(N) = 0 (the case B(x) = 1 was known previously). The complex projective spaces show that this is false in even dimensions. Stable harmonic maps into Hermitian symmetric spaces (6.38)

Here are some further relations between curvature of the target and

stability of a harmonic map of a surface.

Suppose 0: M - (N, h) is a harmonic map from a Riemann surface to a

118

434

J. EELLS AND L. LEMAIRE

Riemannian manifold and 0-'T'(N) is given its Koszul-Malgrange holomorphic structure (4.6). If ¢ is stable and vris /a holomorphic section of O-'T'(N), then

H,(v,v)=-4J (R^(0 ,v) M

(6.39)

Tz

Oz

,v)dx>, 0.

Let N be a compact simple (that is, with simple isometry group)

Hermitian symmetric space. Here is a version of an averaging process of [Lawson, Simons].

If (M, g) is a compact Riemannian manifold and 0: (M, g) -> (N, h, J) a stable harmonic map, then J Trace RN(dcb, v) do = Trace RN(dq, Jv) do

for all v e '(TN). It follows that Im do is J-invariant ; and in particular, evendimensional.

(6.40) Now suppose that d¢ has maximal rank somewhere; then at those points (d,O)`o Jo d¢ is a Hermitian almost complex structure on M. If (M, JM) is a Riemann

surface, then (do)`oJodo = ±JM, so ¢ is either ±holomorphic or do = 0 at isolated points. From (4.4) we conclude that 0 is ±holomorphic on M-and are led to the following theorem of [Burns, Burstall, de Bartolomeis, Rawnsley] and [Ohnita, Udagawa 1]. (6.41) Any stable harmonic map of a compact Riemann surface into a compact simple Hermitian symmetric space is ±holomorphic. Many important special cases preceded the proof of (6.41): the case M = S2, N = P,,(C) [Siu, Yau 2]; the case M = S2, N an irreducible Hermitian symmetric space [Siu 3], [Zhong] (a unified proof has been given by [Burstall, Rawnsley, Salamon], using (6.29)); the case M arbitrary, N = P,,(C) [Burns, de Bartolomeis]. The methods of [Burns, Burstall, de Bartolomeis, Rawnsley] go further, to give the following two theorems. (6.42) Let 0: (M4,g) -* N be a stable harmonic map from a real analytic 4-dimensional manifold to a compact simple Hermitian symmetric space; suppose that rank do > 3 somewhere. Then M has a unique Kahler structure with respect to which 0 is holomorphic. (6.43) If 0: Pm(C) - N is a stable harmonic map into a compact simple Hermitian symmetric space, then is ±holomorphic. In particular, if m > dime N, then 0 is constant. [Ohnita 2] proved (6.43) in case N = P (C); and [Udagawa 3] in case N is a complex quadric and 0 has an additional rank restriction. The first step is to prove that 0 is (1, 1)-geodesic (4.43). In fact, [Ohnita 2] established that for any Riemannian manifold (N, h) a stable harmonic map 0: Pm(C) -> (N, h) is (1, 1)-geodesic. (6.44) In contrast: (1) [Barbasch, Glazebrook, Toth], [Toth 3], and Ohnita have constructed harmonic

non ± holomorphic maps Pm(C) -+ P,,(C) form < n; (2) the Segre twisted embedding P,(C) x Pj(C) - P3(C) is stable harmonic, but not (1, ])-geodesic.

119

ANOTHER REPORT ON HARMONIC MAPS

435

[Udagawa 3] has proved that if 0: (M, g) -- P (C) is a stable (1, 1)-geodesic map of a compact Kahler manifold, then 0 is ±holomorphic (compare (4.47)). (6.45)

See Items Added in Proof, p. 501.

Stable maps of the 2-sphere (6.46) Stable harmonic maps of S2 to Riemannian symmetric spaces are rather well understood, partly due to the following result of [Burstall, Rawnsley, Salamon]. Let 0: S2 -+ G/K be a non-constant stable harmonic map into a Riemannian symmetric space. Then there is a Hermitian symmetric space G,/K, totally geodesically immersed in G/K such that 0 factorises through a holomorphic map W:

S2

' G/K G,/K,.

Moreover, any holomorphic map of a Riemann surface into G,IK, gives a stable harmonic map into G/K. When G/K is irreducible and not Hermitian symmetric, then G,/K, is a complex projective space. (6.47) Compact irreducible semi-simple symmetric spaces G/K can be divided into three classes according to their second homotopy groups. Case n2(G/K) = 0. There are no available Hermitian symmetric subspaces of type G, /K so any stable harmonic map 0: S2 - G/K is constant. Case 7t2(G/K) = 722. In this case both homotopy classes have stable harmonic representatives c: S2 -> G/K, of arbitrarily high energy. Case 7r2(G/K) = Z. These are the Hermitian symmetric spaces. Each homotopy class has a ±holomorphic representative. Any stable harmonic map 0: S2 -' G/K is ±holomorphic (as in (6.41)). (6.48) Using the equality E-index (0) = V -index (0) of (6.36) for harmonic maps 0: S2 -+ G/K, we see that these conclusions are valid for V-stable minimal 2-spheres in G/K.

(6.49) In general, we do not have good estimates for the indices of harmonic maps 0: M -+ G/K. However, here are some special cases. (6.50) If o: S2 S2r (r > 2) is a full harmonic map (in the sense that g(S2) is not contained in an equator of S2r), then [Ejiri 2]

E-index (0) > 2[r(r+ 2) - 3]. (6.51)

Let M be a compact Riemann surface and 0: M -+ S2, a harmonic map

which is real isotropic (in the sense of (7.7) below) and of order s > 1, in the sense that

M, # 0 and M, = 0 for k > s, where M,k is the subset of points of M where the

120

436

J. EELLS AND L. LEMAIRE

dimension of the (k- I)-osculating normal space is 2k. Then the normal bundle V(S2r, M) -+ M is well defined and has a natural complex structure; and V-index (0) > dim, H°(M, V(S2r, M)) + 2r - (2s + 2) [Ferreira 1].

7. Twistor constructions (7.1) There are many different notions of twistor fibration in current usage. We shall say simply that a Riemannian submersion rr: (Z, k, JZ) --+ (N, h) of an almost Hermitian manifold to a Riemannian manifold is a twistor fibration (with twistor space (Z, k, JZ)) if for any almost Hermitian cosymplectic (d *(o = 0) manifold (M, g, J") and holomorphic map V : (M, J") -+ (Z, JZ), the composition

0=noyr:(M,g)-+(N,h)

is harmonic.

The space J(N) (7.2)

Suppose that N is oriented and dim N = 2n. Take Z = J(N), the bundle of complex structures on the tangent spaces of N which are compatible with orientation

and metric. Then jr: J(N) -+ N is a Riemannian fibration associated to the orthonormal frame bundle of N, with fibre the Hermitian symmetric space SO (2n)/U (n).

The vertical tangent bundle T'J(N) has an induced complex structure J'. The horizontal bundle T"J(N) has a complex structure J" via the isometry dn(z) : T" J(N) 9 T (Z,(N),

because z is an almost complex structure on T(Z)(N). We define two almost complex

structures on J(N): (7.3)

JHED

(D

.

These are very different, as indicated by the following properties. (7.4) The structure Jl is a conformal invariant of (N, h) ; and is integrable if and only if (N, h) is conformally flat when n >, 3; or is anti-self dual (that is, the component W, of the Weyl curvature tensor is 0) when n = 2. (7.5)

The structure J2 i

-

r integrable.

Nonetheless, the following property is basic [Eells, Salamon 1, 2], [Salamon 1]: (7.6)

The map n : (J(N ), J2) -+ (N, h) is a twistorfibration.

(7.7)

Let 0: (M, J") -+ (N, h) be a map from a Riemann surface to a real

manifold, and denote by <, >' the bilinear extension of h to Tc(N). The map ¢ is called real isotropic if C = 0 for all a, f > 1, Vz being the covariant

derivative in the direction a/dz. Note that such a map is conformal (the case

a=j7=1).

(7.8) The projection 0 = n o yr of a holomorphic curve yr : (M, J") -+ (AN), J,) is real isotropic, and hence conformal.

121

ANOTHER REPORT ON HARMONIC MAPS

437

(7.9) If (M, J"') is a (1, 2)-symplectic manifold ((dw)1.2 = 0), then the projection 0 = n o w of a holomorphic map yr: (M, JM) (J(N), J2) is (1, 1)-geodesic (in the sense of (4.43)).

We are interested primarily in a converse to (7.6); that is, in when a harmonic map 0: (M, g, JTM) -+ (N, h) is the projection of a holomorphic map (7.10)

w : (M, g, JM) --> (J(N), J2). And (7.9), together with (4.44), indicates that we should restrict our attention to Riemann surface domains, which we do now. (7.11) [Rawnsley 2]. Let M be a compact Riemann surface and 0: M -> (N, h) a map. There is a holomorphic map w : M -. (J(N), J2) with n o yr = 0 if and only if 0 is conformal harmonic and O*w,(N) = 0.

Case dim N = 4 (7.12) When dim N = 4, the special nature of SO (4) offers another presentation of the bundle J(N) as follows. Let A2T(N) and A2T*(N) denote the bundles of 2-vectors and 2-forms on N.

Using their identification via the Riemannian metric h, the Hodge operator determines the spectral decomposition A2T(N) = A+ T(N) e A? T(N),

where At T(N) denotes the bundle of eigenspaces of * with eigenvalue ± 1. Define J±(N) = SAf T(N), the unit sphere bundle of A2, T(N). From the identification of an almost complex structure with its Kahler form (using h) it follows that J+(N) = J(N), whereas J_(N) is equal to the bundle J(N) corresponding to the other choice of orientation on N. These identifications induce the almost complex structures J, and J2 on each of these two spaces. The differential of a conformal immersion 0: M is again a Riemann surface) induces A2dq: A2T(M) --> A2T(N) and thereby Gauss lifts Ot: M-+J±(N) given by 0f = A2do/IA 2dcl (7.13) The map 0: M -+ (N, h) is real isotropic if and only if locally at least one of the lifts q* is J,-holomorphic.

(7.14)

The following is the key to the twistor method for harmonic maps [Eells,

Salamon 1, 2] : The assignment 0 -> * is a bijection between non-constant conformal harmonic maps 0: M -+ (N, h) and non-vertical holomorphic curves * : M -+ (J+(N), J2).

Note that in this statement, 0 is not supposed to be an immersion. In fact, a aj at which do = 0, and the conformal map 0 has only isolated points holomorphic curves ¢' and ¢_ originally defined only on M a standard technique. The above bijections in turn induce one between the non-vertical holomorphic

curves in (J+(N),J2) and (J_(N),J2)-a significant statement, for these are quite different manifolds. (7.15) If (N, h) is Einstein and anti-self dual, then the Kahler form of (J+(N), J2) is (1, 2)-symplectic [Muskarov]. Consequently, every holomorphic map v : M - (J+(N), J2) is harmonic [Eells, Salamon 1, 2].

122

438

J. EELLS AND L. LEMAIRE

If in addition (N, h) has negative scalar curvature, then J+(N) has a symplectic structure compatible with the almost complex structure J2, so results of [Gromov] concerning holomorphic curves are applicable.

(7.16) A study of the holomorphic functions (local and global) on J,(N) has been made by [Davidov, Muskarov] and [Hitchin 1]. In particular, they show that every holomorphic function from (J, (N), J) or (J, (N), J2) to C is constant. The following characterisation is due to [Friedrich]. A map 0: M-+ (N, h) has ¢+ horizontal (that is, dcf, (T(M)) c TH(J, (N))) and holomorphic if and only if for any path (c,),,,,, in M the compositions (7.17)

p iO*(T,(M))

and

(0*(T,(M))1

of parallel translation r along q5(c,) with the indicated orthogonal projections are

conformal.

(7.18) A conformal immersion c: M-->(N,h) is totally umbilic (that (Vdc!)2 0 = 0) if and only if both (7.19) EXAMPLE.

is,

t are J,-holomorphic [Eells, Salamon 2].

Take N = S4 (equivalently, the quaternionic projective line

P,(H) = Sp (2)/Sp (1) x Sp (1)). Then

J+(S4) = SO(5)/U(2) = Sp(2)/U(1) xSp(1) = P3(E) [Bryant 2] calls the real isotropic (7.7) harmonic maps 0: M -+ S4 `superminimal',

and such a map has '+ spin' when t is horizontal. According to [Borel, Hirzebruch], there are four Sp (2)-invariant almost complex structures on J+(S4). They are ±J, and ±J2; J, is the standard Kahler structure. J+(S4) has a 3-symmetric structure arising from J2 [Gray, Wolf]. The Kahler form w of (J4(S4), J2) is (1, 2)-symplectic. The identification of (J+(S4), J,) with P3(C) is at the basis of the Penrose twistor programme. (7.20) EXAMPLE. Take N = R4. Its twistor space is J+(684) = P3(C) - P,(C). A map 0: M - * 684 has + or Vii- horizontal if and only if 0 is holomorphic with respect to some orthogonal complex structure on R. [Micallef 1] has shown that a wide class of oriented V-stable minimal surfaces in 684, including those of finite total curvature, has that isotropy property.

(7.21) EXAMPLE. Take for N a flat torus (T4, h). Then (J+(T4), J,) is integrable but is not (1, 2)-symplectic (this is Blanchard's variety [Atiyah, Hitchin, Singer]), whereas (J+(T4), J2) is (1, 2)-symplectic but not integrable.

(7.22) EXAMPLE [Eells, Salamon 2].

Let (N, h) be ± self dual and Einstein;

denote by ± the set of horizontal holomorphic curves M-*J+(N), and ' the equivalence relation that identifies two curves in U .x.02_ having the same projection 0 in N. If the curves are distinct, then one is c+ and the other O-; in that case 0 is totally geodesic. There is a bijective correspondence between real isotropic harmonic maps 0: M -.' (N, h) and the set (,Cat, U

123

ANOTHER REPORT ON HARMONIC MAPS

439

(7.23) EXAMPLE [Fells, Salamon 2], [Gauduchon]. The real isotropic harmonic maps 0: M - P2(C) are ± holomorphic, or 8-Gauss maps of holomorphic curves (in the sense of (8.26) -(8.33) below), or totally real.

(7.24) A map 0: (M, JM) -- (N, h) of a Riemann surface to a Kahler manifold is called complex isotropic if = 0 for all a,3 > 1. (7.25)

[Eells, Salamon 2]. A conformal harmonic non ±holomorphic map

M (N, h) into a Kdhler surface is complex isotropic if and only if ¢_ is horizontal. (7.26) Let ¢: M2 -* S' be a conformal branched immersion. Twistor conditions (involving harmonicity or holomorphicity of the Gauss lifts ¢+) to ensure that the tension field r(q) be holomorphic or parallel are given in [Ejiri 3] and [Ferreira 2].

The space Q(N) (7.27) For higher dimensional ranges, partial twistor spaces have been studied systematically in [Rawnsley 2]. See for instance [Fells, Salamon 1], [Eells 5]. Let (N, h) be an oriented n-manifold and n: Q(N) -+ N the bundle of oriented 2SO (n)/SO (2) x SO (n-2). subspaces. Its fibre model is the complex quadric

Each point q e Q(N) represents an oriented Euclidean 2-space in T,(q,(N), and therefore a complex line Lq. We define a subbundle IT -> Q(N) of the tangent bundle of Q(N) as follows. The fibre 11q is the subspace of T(Q(N)) spanned by the lift of Lq to the horizontal subspace TQ (Q(N)), and the vertical subspace TQ (Q(N)). These

components of IT, have complex structures JQ and J', resp. Consequently we can define two complex structures on the bundle IT -- (N, h) by Jl

(7.28)

dyi(T(M))

(J" on T"(Q(N)) n ri 1J`' on T`(Q(N))

J" on T"(Q(N)) n II J2

1-J' on T'(Q(N)).

We say that a map yr: (M, J) - Q(N) is JJ- or J2 holomorphic if II and dip o J = Jl o dp or dyr o J = J2 o dyi. We say that it is horizontal if

dyr(T(M)) c T"(Q(N)). A conformal map yr : M -+ Q(N) is horizontal if and only if it is both J1- and J2holomorphic. In that case we call yr horizontal holomorphic.

Let 0: M -+ (N, h) be a conformal harmonic map of a Riemann surface ; : M -+ Q(N) its Gauss lift, assigning to each x e M the image

(7.29)

and

do(x) TX(M) e Q4,(T)(N)

if

d¢(x) 0 0;

and extending over the zeros as in (7.14). Then [Fells, Salamon 1]: The assignment g -+

is a bijection between non-constant conformal harmonic maps

0: M - (N, h) and non-vertical holomorphic maps : M - (Q(N),.12).

124

440

J. EELLS AND L. LEMAIRE

(7.30) EXAMPLE. Take N = R'. Then associated to a conformal harmonic map 0: M --+ Rn (that is, a minimal branched immersion of M in R") is the diagram

Q(Rn) =

Rn

X Qn-2 --> Qn-2

0

the indicated projection p being antiholomorphic as a map (Q(l ), J2) -+ Qn-2. Thus we obtain a theorem of [Chern 2]: (7.31) The Gauss map y,, = p o c : M--+ Qn-2 is antiholomorphic if and only if the conformal map 0: M -+ R' is harmonic.

(7.32) EXAMPLE [Eells, Salamon 2], [Eells 5]. Take n = 3. Then Q(N) is the unit sphere bundle ST(N) of N. The J J2 structures are examples of almost CR-structures

on N (4.48). The structure J, is always integrable [Lebrun]. If 0: M -+ (N, h) is a conformal map and dd(x) 96 0, we identify q(x) e ST(N) with the positive unit vector orthogonal to do(x) T(M). The second fundamental form fl(c) is identified with do. Say that 0 is pseudo-umbilic if <,8(0), r(q)> = tg for some smooth function

a.: M - R. (7.33) The correspondence 0 -> q is a bijection between non-constant conformal harmonic maps 0: M-+ (N, h) and non-vertical holomorphic maps c: M (ST(N), J2). (7.34) A conformal map 0 is totally umbilic (7.18) if and only if is J,holomorphic. The map 0 is pseudo-umbilic if and only if q5 is conformal [Obata] ; and is totally geodesic if and only if is horizontal.

Twistor bundles over symmetric spaces (7.35) Our aim now is to obtain finer twistor constructions in case (N, h) is a symmetric space.

Let G be a compact semi-simple group and P a parabolic subgroup of the complexification GC (that is, P contains a maximal solvable subgroup of GI). Then G

acts transitively on the homogeneous space G'/P, so that Gc/P : G/H where H =P n G is the centraliser of a torus in G; in particular G/H is a flag manifold. Now GC/P is a complex manifold, so that the above identification, together with

the metric on G/H induced from the Killing form on G, endows G/H with an integrable Kahler complex structure JP,. Choice of P also determines the canonical element crP in the centre of the Lie algebra L(H) for which ad ,, has eigenvalue \/(-1) . r on the rth step of the descending central series of the nilradical of L(P). Then Ad exp 1r p is an involutive automorphism of L(G), so that if K,, is the identity component of its fixed set, then H C K,,. The homogeneous fibration (7.36)

np: G/H -+ G/Kp

fibres a flag manifold over a symmetric space; in fact, G/Kp is inner symmetric (that is, rank G = rank K,.).

125

ANOTHER REPORT ON HARMONIC MAPS

441

Define the almost complex structure JZ on G/H by reversing the orientation of J; on Ker dnP. Again, J2 is never integrable. However [Burstall, Rawnsley 1, 31: (7.37) The map (7.36) is a twistor fibration. Every compact inner symmetric space has such a flag twistor fibration arising by the above construction. (7.38) If 0: S2 --* G/K is a harmonic map to an inner symmetric space, there is a parabolic subgroup P such that K = KP and a holomorphic map w : S'--+ (G/H, J2) such that 71, o w = [Burstall, Rawnsley 1, 3]. (7.39)

Analogously, Burstall and Rawnsley prove that if : M -* G/K is a

harmonic map of a compact Riemann surface of genus p into a Hermitian symmetric space, with ramification index r(O) > 2p-1, then again there is a parabolic subgroup P with K = KP and a holomorphic map w: M --+ (G/H, J2) with nr,o W = . (7.40) Here are the main ideas in the proof of (7.38). Use the moment map (see (8.45) below) of T(G/K) to view it as a subbundle of the trivial bundle (G/K) x L(G). Treating do as a L(G)-valued 1-form on S2, we equip the trivial bundle S2 x GC with

the G-connection V = d-dd. Then 0 is harmonic if and only if d¢(aa,) is a local holomorphic section with respect to the Koszul-Malgrange structure on (S2 x Gc, V O).

Use the Birkhoff-Grothendieck classification theorem (4.9) to reduce that to a Pbundle for some parabolic subgroup P, and hence obtain a map w: S2 --+ G/H = Gc/P

with H c K which is J2-holomorphic in the vertical directions. A direct analysis shows that w is J2-holomorphic in the horizontal directions as well. The case of domains M of higher genus uses the Harder-Narasimhan filtration of a holomorphic vector bundle on M, in place of the theorem of Birkhoff and Grothendieck. (7.41) EXAMPLES.

If G/K = S2i, then there is just one such flag twistor space :

SO (2n + 1)/U (n).

If G/K = P (E), the only such are SU (n+ 1)/S(U (r) x U (1) x U (n-r)), I < r < n-1. In these examples the Kahler form co for JZ is (1, 2)-symplectic. (7.42)

The twistor spaces over a symmetric space G/K with integrable J, and

holomorphic horizontal distributions have been classified [Bryant 4] and [Salamon 1, 2]. They all have the form (7.36) with (1, 2)-symplectic co'. =

The space G,(T'(N)), complex isotropy (7.43)

Thinking of a complex line in C' as an oriented plane in F12' defines a

totally geodesic embedding of P,-,(C) in Q2ri_2. Thus for a Kahler manifold (N, h) we have the bundle inclusion

G,(T'(N)) '

Q(N),

N

Z

126

442

J. EELLS AND L. LEMAIRE

where G,(T'(N)) denotes the Grassmann bundle of complex lines in T'(N). Now let 0: M --> (N, h) be a conformal harmonic map of a Riemann surface; and assume that 0 is not antiholomorphic; define yr : M -> G,(T'(N)) by taking yr = span 8¢(0,) at points where do A 0; and extending over the zeros as usual. Then: (7.44)

The correspondence 0 -+ yr is a bijection between the conformal harmonic

maps 0: M --+ (N, h) which are not antiholomorphic and the holomorphic maps yr: M -- (G,(T'(N)), J2) not projecting to an antiholomorphic map. (7.45) More generally, the Grassmann bundle G,(T'(N)) -. N of r-planes in T'(N) of a Kahler manifold (N, h) is a twistor fibration.

Note that G,(T'(N)) is a subbundle of J(N). J, is integrable on G,(T'(N)) if and only if the Bochner curvature tensor of (N, h) vanishes [O'Brian, Rawnsley]. If.v: M -+ (G,(T'(N)), J,) is horizontal and holomorphic, then it o yi = 0: M -> (N, h) is complex isotropic (in the sense of (7.24)) [Rawnsley 2]. In general, it is not easy to determine which 0 are such projections. A suitable condition has been given in terms of curvature [Rawnsley 2]. (7.46) ExAMFL.E. The description of harmonic maps of surfaces into spheres, projective spaces and Grassmannians ((8.37) below) can be interpreted in terms of twistor constructions as follows. Let f: M P (C) be holomorphic and full (that is, f(M) lies in no proper projective subspace); its rth associated map f,: M --+ G,,-,(C) is defined in terms of a local lift f,: U -+ Cs+1\0 off, where U is an isothermal chart of M, by

.fr(x) = Span {fu(x), 0 fu(x), ..., azfu(x)},

again after natural extension. Now U (n+ 1)/U (r) x U (s) x U (1), with r+s = n; we can view its points as pairs {(V, W): VeG,(Cxi+1) WeG,(C"+1), V1 W}.

Let yr = (f,_ f,): M-+ G,(T'(P (C))); that gives a bijection between full holomorphic maps f and full horizontal holomorphic maps yr. Now define ¢ = n o yi : M -+

so fi(x) =f,, ,(x)11 f,(x)(= G'(f)(x) in the notation of (8.37) below). Then [Eells, Wood 4] : (7.47)

The assignment yr - 0 is a bijection between the full horizontal holomorphic

maps yr: M -+ (G,(T'(P (C)), J,)) and the full complex isotropic harmonic maps 0: M -+ P (C) with order r: max dim Span{V= 0(x) :

1

a} = r.

ZEM

When M = S2, every harmonic map to (7.48)

is complex isotropic.

Combining the above bijections gives the following form [Eells, Wood 4].

The assignment f -+ 0 = ff, n f, is a bijection between pairs (f, r), where f is a full holomorphic map and 0 ` r < n, the full complex isotropic harmonic maps 0. (7.49)

Calabi's theorem (8.40) takes the form :

127

ANOTHER REPORT ON HARMONIC MAPS

443

Consider the subbundle (n = 2r) A, = SO (2r + 1)/U (r) of G,(T'(P2r(C)) over the real projective space P2r(R) c &(C). Then : There is a bijection between full horizontal holomorphic maps yr: M -- (Jrr, J,) and full real isotropic harmonic maps 0: M -- P2r(R) (7.50) [Gauduchon, Lawson]. Any injective harmonic map 0: S2 _ S5 is totally geodesic. By way of contrast, there are minimal embeddings S2 --+ Se by spherical harmonics, which are not totally geodesic. (7.51)

Similarly, we define for n = r+s+t,

r8C=U(r+s+t)/U(r)x U(s)x U(t). Then, with their definitions [Erdem, Wood]: There is a bijection between full horizontal holomorphic maps v: M

and full

strongly isotropic harmonic maps 0: M -- G, (C") of order r. For r > I not every harmonic map 0: S2 - Gr(C") is isotropic. (7.52) Similar parametrisations of isotropic harmonic maps of Riemann surfaces into other symmetric spaces can be found in [Erdem, Glazebrook], [Glazebrook 1, 2].

Twistor degrees (7.53)

Let (N, h) be an oriented 4-manifold. Then the first Chern class

c,(N, J2) = 0. If M is a compact Riemann surface and 0: M -- (N, h) a non-constant conformal harmonic map, its twistor degrees dt(¢) are defined [Eells, Salamon 2] by (7.54)

2df(o) = ct

cl(V1.o)[M],

where V' 0 denotes the bundle of (1, 0)-vectors relative to J tangent to the fibres of J± (N). (7.55) The degree d f(q) is an integer whenever N is a spin manifold (that is, its second Stiefel-Whitney class vanishes). In this case the pullbacks ¢-let of the spin bundles A± of N contain holomorphic line bundles of degrees dt(¢) [Salamon 3].

(7.56)

For any conformal harmonic map 0: M --> (N, h),

2dt(0) = X(M)+r(o)±e, where X(M) is the Euler characteristic of M, r(o) the ramification index of ¢, and e the Euler number of the normal bundle of c(M) in N. In particular, d+(O) + d-(O) = X(M) + r(o),

d+(O) - d-(0) = e.

Assume also that N is compact. If 0: M (N, h) has no branch point and all self intersections are transverse, then (7.57)

e = [O(M)] o [r6(M)]-so,

where o denotes homological intersection in H2(N,ZL); and s. is the geometrical selfintersection index, counting with multiplicities the points of M on fibres of cardinality 2.

128

444

J. FELLS AND L. LEMAIRE

(7.58) EXAMPLE [Friedrich], [Poop]. Let (N, h) be self-dual and Einstein, and V(¢) the area of the map ¢: M-- (N, h). If the lift q_ is horizontal holomorphic (so that 0 is real isotropic), then

d_ = 48rc V*

Scaly"."

The twistor degrees of a conformal harmonic map ¢: M - S' are the ordinary Brouwer degrees of the lifts in P3(C): d+(0) = deggt. (7.59) EXAMPLE.

(7.60) If 0 is real isotropic but not totally geodesic, then lel > 2p+4-r, where genus M = p. If 0 is not real isotropic, then -3(p- 1) <, deg p-1.

The case p = 0 is due to [Ruh]: any minimal immersion of S2 in S' with trivial normal bundle is totally geodesic. By refining the arguments in [Eells, Salamon 2], [Salamon 3] has shown that a harmonic map 0: S2 - S' is totally geodesic if and only if e = 0. And for 0 not totally geodesic, lel > 4+r [Gauduchon]. (7.61)

[Bryant 2] has obtained a Weierstrass formula for real isotropic harmonic

maps from a Riemann surface M to S. Indeed, he has shown that for any pair of meromorphic functions F and G on M with G non-constant, the map yr: M - P3(C) given by

dF

1 dF

yr = [1, F-2GdG'G'2

dG

is horizontal holomorphic, and that every horizontal holomorphic map whose image is not contained in a P1(C) has that form. (7.62)

A theorem of Hirsch and Smale [Hirsch] asserts that the regular

homotopy classes of immersions of S2 in S' (that is, immersions homotopic through immersions) are parametrised by the Euler number e of the normal bundle; and all even e are realised. We conclude that the class e = 2 has no harmonic representative. (7.63) EXAMPLE [Salamon 3]. Let 0: M -* (N, h) be a conformal harmonic map of a Riemann surface M into a Kahler surface (N, h), and set c(¢) = c1(O-1T(N)) [M]. An examination of the induced holomorphic structure on the pullback 0-1A2 T*(N) of the bundle of self-dual 2-forms shows readily that

2d+(o) =X(M)+r(c)+e

Ic(o)I < 0,

unless 0 is ±holomorphic with 2d+(¢) = Ic(&)I > 0. When (N, h) = P2(C), we have c(c) = 3 deg ¢. It follows from (7.57) that any embedded minimal 2-sphere in P2(C), is either a complex line or a conic. That was first proved by [Eschenburg, Guadalupe, Tribuzy] and [Webster]; and subsequently generalised by [Gauduchon] who proved that the same assertion is valid for any injective minimal 2-sphere. See [Gauduchon, Lawson], where it is also proved that an isotropic minimal embedding of a closed Riemann surface in P2(C) is ±holomorphic. (7.64) EXAMPLE.

Consider the isotopy classes of maps from S2 to a K3 surface

N. In each homotopy class of maps of ir2(N) = 7122, the isotopy classes are parametrised by their normal class, taking all even values [Hirsch]. Because cl(N) = 0,

129

ANOTHER REPORT ON HARMONIC MAPS

445

we get c(ql) = 0 and (7.63) implies that for a harmonic immersion 0: S 2 -+ N, e 5 - 2.

Therefore, many isotopy classes of maps S2 -+ N have no harmonic representative. (7.65) EXAMPLE [Eells, Salamon 2].

The flag manifold

F= U(3)/U(1) x U(1) x U(1) is the twistor space of P2(C). It carries 8 invariant almost complex structures, exactly

6 of which are integrable [Borel, Hirzebruch]. Let J be the unique (up to sign) non-integrable one. Denote by ;r,: F -+ P2(C) the three homogeneous projections (0 i < 2). If yy : M -+ (F, J) is holomorphic, then each of the three compositions 0, = n, o yr: M -+ P2(C) is conformal harmonic. Conversely, starting with a conformal harmonic map 0u : M --+ P2(C), its Gauss lift yr = co_ via no is J-holomorphic ; so 01, 02 are also conformal harmonic. (7.66)

If o: M -+ P2(C) is conformal harmonic, then [Eells, Salamon 2]

2 deg(¢1) = -deg(0o)-2d_(g0); 2 deg (02) = - deg (00) + 2d_(gio).

If 0o is not complex isotropic, then [Eells, Wood 4]

Ideg(O.)I < 2p-2. Strings (7.67)

Let M be an oriented surface with nondegenerate quadratic form

9= g(l.l) of signature (1, 1); and N an oriented n-manifold with quadratic form h = h(",4) of signature (p,q) with p+q = n. A string is a conformal harmonic map :(M,g)--+(N,h). Relative to an isothermal chart on (Mg), conformality is expressed by (7.68)

I0u12 + 10v12 = 0 = (0., Ov> ,

and harmonicity by the hyperbolic system (1 < y 5 n) : (7.69)

ouu-7

LV

v V) = 0.

Those are the Euler-Lagrange equations of E(O) = if, (1¢ul2 - I0v12) vg. z

(7.70) EXAMPLE. A map 0: l1.1 -+ R".4 is a string if and only if there are null ' (that is, la'(t)120 = IP'(t)12) such that

paths a,fl: R -+ IJ

O(u, v) = a(u-v)+fl(u+v). (7.71) EXAMPLE.

Take (M, g) = l1.' and (N, h) = S(Rp' 4) = (y e RP, 4: I y12 = 1),

with induced h. This is a homogeneous hypersurface of constant curvature + 1. Conformal harmonic maps 0: l1.1 -+ S(EFV"') have been studied by [Gu 3, 4] and [T. K. Milnor 1, 2] in the case (p, q) _ (2, 1); their solutions are given explicitly via the sinh-Gordon equation au. - a,,,, = - sinh a. [Erdem] and [Ejiri 5] have studied isotropic harmonic maps of Riemann surfaces S(ln-1.1) into

130

446

J. EELLS AND L. LEMAIRE

(7.72) For p < r, q < s consider the natural inclusion RP. ? -* W..'. If 0 (r, s) we have the Grassmannian denotes the orthogonal group of SO (r, s) GP. v(r, s) = SO (p, q) x SO (r -p, s -

q)

of oriented subspaces of signature (p, q) in D '. That is a symmetric space; and an open orbit in the dual of GP+Q,,+a [J. Wolf], with

pseudo-Riemannian structure. The Grassmannian G,.,(r,s) is pseudo-Kahler. Its tangent bundle T(G,.,(r, s)) = L* (& L' has a canonical decomposition of the form T(G,,,(r, s)) = (L+ (9 W) e (L_ ® W). There is an almost product structure (a real endomorphism J with J2 = +Id) having L. as + 1-eigenspace. (7.73) With every immersed string 0: (M, g) -+ R', we have its Gauss map yo: (M, g) - G,. ,(r, s). The decomposition T(M) = K+ e K_ is determined by the natural almost product structure JM on T(M); and 0 is harmonic if and only if dyo

preserves types. (7.74)

Now suppose p = 2 = q, so dim N = 4. The Grassmannian bundle

G,.,(N) --> N of subspaces of signature (1, 1) has fibre the Lorentz hyperbolic plane S1, 1, the space of null 2-planes in R2'2. As in (7.3), the space G,. ,(N) has two almost product structures J,, J2 (with J, integrable if and only if (N, h) is anti-self dual; and

J2 never integrable). The argument of (7.14) gave Eells and Salamon the following (preliminary) analogue. (7.75) There is a bijective correspondence between conformal harmonic immersions : (M, g11,1)) -> (N, h".2)) and type preserving maps 0: (M, g«. 1I) -- (G1 1(N), J2) every-

where transverse to the fibres. (7.76) REMARK. The analogue does not extend to the case (p, q) = (3, 1), because the group SO0(3, 1) is simple (in contrast to SO (4) and SO,(2, 2)).

Harmonic morphisms

Everywhere in the theory we see that harmonic maps of Riemann surfaces to manifolds have special properties, related to dimension two. On the other hand, harmonic morphisms (in the sense of (2.30)) are special when the dimension of the range is two (see, for example, (4.5) and (2.34)). This was further confirmed by [J. C. (7.77)

Wood 3, 4], who formulated in particular the following twistor description of harmonic morphisms. (7.78)

Let (Mg) be an oriented 4-manifold and N a Riemann surface. Let

¢: (M, g) --+ N be a harmonic morphism which is a submersion everywhere. As in

(2.30), denote by T°(M) and T'(M) the associated vertical and horizontal distributions. The orientations of M and N induce orientations on both distributions. For these orientations, rotation of n/2 in the planes induces complex structures J,, and JH on the distributions, and one can define two almost complex structures

J, = (J,,,JH) and J2 = (Jr, -Jt,) on M. Note that the first is compatible with orientation of M, and the second not.

131

ANOTHER REPORT ON HARMONIC MAPS

447

(7.79) Now we have seen in (7.12) that J+(M) (resp. J_(M)) is the bundle of almost complex structures on M compatible (resp. incompatible) with the orientation, so that we have defined two maps y,: M -+ J+(M) by x -+ J,(x), and y,: M - J_(M) by x --+ J2(x).

(7.80) [J. C. Wood 3, 4]. The association -+ (y1, y2) induces an injection from equivalence classes of submersive harmonic morphisms from M to a Riemann surface, to smooth sections (y1, y2) of J+(M) x J_(M), where y, is J2-holomorphic and y2 is J,-antiholomorphic. 8. Maps into groups and Grassmannians (8.1)

The aim of the twistor programme of Section 7 is to describe the harmonic

maps from a surface to a manifold in terms of holomorphic maps. However, in general their range is an almost complex manifold with non-integrable structure. The

primary objective of the present Section is to show how, in certain situations, harmonic maps can be built out of holomorphic ones with a complex manifold as range.

The origin of these ideas lies in the papers of [Calabi 1, 2], which appeared at about the same time as the start of Penrose's twistor programme. His main theorem is presented in (8.40) below as a special case of its subsequent developments.

The ideas leading to the factorisation theorem (8.19) are due to [Zakharov, Mikhailov], [Zakharov, Shabat], and [Uhlenbeck 5]. Maps to a Lie group

Let G be a compact Lie group endowed with a bi-invariant metric. Its connection is given by V, Y = 21X, Y], and its curvature by R(X, Y) Z = 4[[X, Y], Z]. (By way of contrast, the connection V, Y = [X, Y] has (8.2)

Levi-Civita

torsion T(X, Y) = [X, Y] and curvature 0.) (8.3)

Let L(G) denote the Lie algebra of G. The Maurer-Cartan form µ of

G is the L(G)-valued 1-form on G given by p(v) = v for all v e L(G) (seen as a left invariant vector field on G). We shall write [p AU] for the 2-form given by [ft A p] (X, Y) = 2[p(X), p(Y)]. Then dp+2[p A p] = 0. The Jacobi identity ensures that d(p A [p A p]) = 0. We can view p as a connection 1-form on T(G) with curvature 0.

If 0: M - G is a map of a Riemann surface to G, we denote by a = 0*,u the pull-back of the Maurer-Cartan form. It is an L(G)-valued 1-form on M with da+2[a A a] = 0. Then a = ¢-ldq. The complexification of a-still denoted by a---can be split into complex types: a = a'+a ", where a" = X. Then da and 7a'+ 7a"+ [a" A a'] = 0.

(8.4)

Now *a = - ia' + ia" (where * is the Hodge operator), and we have (see (8.46) below) :

(8.5)

A map 0: M--+G is harmonic if and only if d*(qS*p) = 0; equivalently:

(8.6)

Da' = 80".

Applying (8.4), we find (8.7)

aa'+2[a" n a] = 0;

taking conjugates we see that (8.4) and (8.6) together are equivalent to (8.7).

132

448

J. EELLS AND L. LEMAIRE

(8.8) Here is an application of (8.5), due to Uhlenbeck. A non-constant harmonic map 0: S2 --> G is unstable. (That also follows from (6.47), treating G as a symmetric space.) Indeed, *c*µ is a closed L(G)-valued 1-form on S2, and so there is a function

: S2 -* L(G) with do = *q*,u. Then can be viewed as a variation of 0; and its associated deformation (0) is energy-decreasing:

T (8.9)

Writing or' = 2Az dz and of = 2Az dz in a complex chart of M expresses (8.6)

and (8.7) in the equivalent forms 7A, + 8Az = 0,

Note also that E(q) =

'

0A2 + [AZ, AZ] = 0 = 8A= - [Az, AZ].

trace A, Az dz A dz.

L112dx = - 2i JM

(8.10) Let L = 8+(1-A`) A, and K = -(1-)..)A2 for any AeC*. Valli has shown that if 0: M -* G is harmonic, then L and K satisfy the Lax equation

8L/8z = [K, L]. (8.11) In particular, the spectrum of the problem Lip = Ayr subject to the condition ayi = Kyl varies holomorphically.

[Hitchin 2] used this formalism-emphasising its gauge interpretation-to study harmonic maps from the torus T2 -+ SU (2) = S3 (a situation in which the methods of Section 7 are not effective). He reduced their study to algebraic geometric questions involving a certain hyperelliptic curve C, called the spectral curve of the (8.12)

map. Thus new harmonic maps 0: T2 -+ SU (2) are obtained as well as certain deformations of these; see also [Toth 5]. As special cases: genus C = 0. If 0 is conformal harmonic, then it is a finite covering of the Clifford

torus. If 0 is harmonic but not conformal, it is a finite covering of a rectangular torus. genus C = 1 includes the Gauss map of Delaunay's surface [Eells, Lemaire 2] ; and [Hsiang, Lawson]'s minimal tori in S3 invariant under a circle action. genus C = 3 includes the Gauss map of [Wente 1, 2]'s immersions. (8.13)

Let M be a compact Riemann surface, and p: n1(M) --* PSL(C2) = SL (C2)/centre

an irreducible representation (in the sense that p(n1(M)) fixes no point of P1(C)). With induced action of n1(M) on hyperbolic 3-space Q8H3 (via the description PSL (C2)/SO (3)) we form the associated flat bundle n: W = M x P IRH3 -- M, where M is the universal cover of M; the fibres are isometric to RH3. The existence theorem (3.33) has been extended to sections of Riemannian fibration [C. M. Wood 1, 2]-with growth restrictions if the fibres are non-compact. In the present context that has been sufficient for [Donaldson 3] to obtain harmonic sections of it, and hence solutions to the system of equations (8.14) below. For any section s of it he considers the principal SO (3)-bundle P. - M associated to the pullback s 1(T'(W)) of the vertical tangent bundle of W, with induced SO (3)-connection

133

ANOTHER REPORT ON HARMONIC MAPS

449

V,; that is flat in the horizontal directions and restricts to the Poincare metric on the fibres of W. Let a, = 2i(d''s), where d°s is viewed as an LC(P,)-valued 1-form on M. Then

R,+

(8.14)

0;

d,

0,

d*a=0

if and only ifs is a harmonic section of n. Here R, denotes the curvature form of the connection V,; d, is the exterior differential on the appropriate vector bundle valued forms, using V,; and d* its adjoint operator.

These equations are equivalent to the self-duality equations introduced by [Hitchin 3], who has examined them thoroughly-especially with respect to matters involving gauge equivalence and stability of vector bundles over M. [Valli 3] has applied the energy-reduction method of (8.19) below to solve a similar equation in inhomogeneous form, for suitable connections.

A factorisation theorem (8.15)

Take G = U (n), the unitary group, whose Lie algebra consists of the

skew-Hermitian n x n matrices, endowed with the bi-invariant metric which on L(G) is = Trace uv*, where v* = U. (8.16) The involution B - B-' on U (n) has fixed set {Be U (n) : B2 =I); that is identified with the Grassmannian Grass (C") of complex vector subspaces of C" via the correspondence P - P - P1 = B e U (n), for each Hermitian projection P onto the subspace Im P e Grass (C"). Here Grass (C") = l f ;`_0 G,(C"), where Gr(CI) is the complex Grassmannian of r-

planes in C. Each Gr(C") is a Kahler manifold (an irreducible Hermitian symmetric space), totally geodesically embedded in U (n). By that embedding, G,(C") is identified with {BeU (n): B2 = 1 and its (+ 1)-eigenspace is r-dimensional}. (8.17) To each map 0: M - Gr(C") c Grass (C") of a Riemann surface M, we associate the vector subbundle 0 c MX C" of rank r, where for each x e M, the fibre is the point q(x) E Gr(C" ). The correspondence 0 -+ 0 is bijective. The bundle 01 is associated to the map - 0 (where 0 and - ¢ are seen as maps from M to U (n), using the above embedding). The map 0 = P- P1 is holomorphic if and only if P1 DP = 0, or equivalently if the bundle P is a holomorphic subbundle of M x C. (8.18) The following proposition is due to [Uhlenbeck 5], refined by [Valli 1]. Let M be a compact Riemann surface and 0: M --# U (n) a harmonic map. Then for P: M --> Grass (C") satisfying P'AZ P = 0 and Pl(0P+Az P) = 0, the map = ¢ - (P-P1) : M --> U (n) is harmonic. Furthermore, E(0) = Area (M) c,(P) [M], where c,(P) is the first Chern

class of P. In case M = S2 and 0 is non-constant, P can be chosen so that E(¢) - E(0) < 0. That requires use of the Birkhoff-Grothendieck theorem (4.9) on the structure of

holomorphic vector bundles over S2; indeed, from that theorem we obtain a canonical choice of P to minimise 15

E(li). BLM 20

134

450

J. EELLS AND L. LEMAIRE

(8.19) An immediate consequence is [Valli 1]'s version of a theorem of [Uhlenbeck 51: Associated to each harmonic map 0: S2 -+ U (n) is a sequence o, ... , 0k of harmonic maps S2 --+ U (n), with 0, constant,

Y'k-0,

0j=¢j-,(P-P1) (I <j 4n.

Thus we have the canonical factorisation 0 = 0o(Pi -- Pi) ... (Pk - Pk ). Each factor is holomorphic with respect to a specific connection.

We observe that E(0) = -4rz E'-, cl(P,). Thus the energy of any harmonic map 0: S2 U (n) is an integral multiple of 4,t. In fact, a similar integrality theorem is valid for arbitrary compact groups. (8.20) [Burstall, Rawnsley 1] have established the following generalisation of (8.19). Let G be any compact simple group 0 G2, F4, E8. Define r = {c c- L(G) : exp 2nc = e}

and y: F --+ G by y(c) = exp irc. If 0: S2 --+ G is harmonic, then there exist (i) (ii) (iii)

i < k), Hermitian symmetric spaces H, c IF, maps V,: S2 -+ H, harmonic maps fit: S2 --+ G (0

such that 0o is constant, ¢k = 0, and 0, =/,¢i_,(y o yi1j'. Thus we have the decomposition

0 = 00(Yo P1)...(YDY'k) Furthermore (compare (9.39)), k < E(¢)/ 16n. (8.21) REMARK.

(8.19) is just one of the many factorisations. In the case of

harmonic maps 0: S2 -' U (n) we have just obtained Oj_, from 0, by reducing energy. [Uhlenbeck 5] reduces her uniton number. [J. C. Wood 6, 7] uses a method analogous

to that of [Wolfson 3]. In the case of harmonic maps ¢: S2 -+ G, (C") c U (n), [Burstall, Salamon] reduce the length of their Birkhoff-Grothendieck decomposition; [Wolfson 3] factors by means of a-Gauss bundles G-`(¢) (see (8.26) below). Other factorisations have been given in special cases: [Eells, Wood 4] for r = 1; [Chern, Wolfson 3] for r = 2; [Burstall, Wood] for 2 < r < 5; and [Erdem, Wood] for strongly isotropic maps. Maps to complex Grassmannians (8.22) The above factorisation can be further refined and made more explicit in case the target manifold is a complex Grassmannian. We now describe this case in more detail. The presentation here was provided by Burstall, as preferable to one emphasising [Burstall, Salamon]. (8.23)

Let 0: M--+G,(C") be a map of a Riemann surface into a complex

Grassmannian, and 0 the corresponding rank r subbundle of M x C" (8.17). Then there is an identification 0-1T1,0(G,(C")) - 0* ® #1

135

451

ANOTHER REPORT ON HARMONIC MAPS

under which the (1,0)-part of do corresponds to the second fundamental form of

4 c M x C. We may write the components of this second fundamental form locally

A'= n'oOon, A;= nloaon,

where n: M X C" --+

is the bundle projection and 3, a are the flat differentiations in

M x C. These induce connections in 0 and 0'-, and thus Koszul-Malgrange holomorphic structures (4.6). A map 0 is harmonic if and only if the second fundamental form A; is holomorphic; that is, A o a. = a,, o A;, where a, (resp. a,l) denotes the a-operator on 0 (resp. 01). (8.24)

A special case of (8.18) in our situation is the following result ([Burstall,

Salamon]).

Let 0: M --> G,(C") be a harmonic map and let a c , f c 1 be holomorphic subbundles such that A;(a) c fi and A',(#) c at. Then the map _ (on al) Q is harmonic. Furthermore,

EO - E(¢)

(8.25)

4mc1(a (D ft).

In the terminology of [Burstall, Wood], ¢ is obtained from 0 by forward replacement of a by ft. There is a dual notion of backward replacement of antiholomorphic subbundles preserved by A", 0 being obtained from by backward replacement of iq by a. Such methods for producing new harmonic maps from old ones have been used by [Burstall, Wood], [Burstall, Salamon], and (Chern, Wolfson 2, 3]. (8.26)

If 0: M -+ Gr(C") is harmonic, there is a holomorphic subbundle of 01

which coincides with the image of A; almost everywhere. We call it the a-Gauss bundle

and denote it by G'(0). Similarly, A; defines the (anti-holomorphic) a-Gauss bundle G""(0).

Taking a = , f = G'(0) in (8.24) shows that G'(0) (and similarly G"(95))

correspond to harmonic maps into a Grassmannian G,(C") with t 5 r. Thus we may iterate the procedure, defining bundles and hence harmonic maps by G'(G" 11(x)),

G1(0) = G'(0)

G°(G c8

G`

Gc-"(g5) = G"(O)

s>1

s > 1.

The sequence of harmonic maps G(`)(0) together with their second fundamental forms Az = Acm(,) is called a harmonic sequence in [Wolfson 3]. (8.27)

Suppose now that each A{ is an isomorphism on almost all fibres : A,

Ao

G'(0)

G`2>(§6) ---f ... ---> G`8>(0).

Then we have:

(8.28)

c1(G">(0)) =

c1(Gca-1)(0)) +r(2-2p)

+ram (Ai-1),

where p = genus (M) and ram (A-1) is the number of zeros of dz Qx A'1 counted with their multiplicities. From this, [Wolfson 3] obtains the estimate (8 . 29)

(s + 1) c l (0) + r(2 - 2p )

s(s + 1)

2

8

+ Es ram (A , ) 5 E(0 ) :-o >5-z

136

J. EELLS AND L. LEMAIRE

452

and so concludes: (8.30) which

For any harmonic map 0: S2

G, (C") there is some G11>(0) with i > 0, for

rank G"+"(0) < rank Gl')(O);

and thus by iteration the harmonic sequence must terminate; that is, there is a q for which G(Q)(qS) = 0,

G(4+1)(q) = 0.

(In the terminology of [J. C. Wood 6], such maps have finite 8-order.)

(8.30) may be interpreted as giving a very explicit form of Uhlenbeck's factorisation (8.19). [J. C. Wood 6, 7] has provided a proof of (8.30) using (8.28) together with the energy formula (8.25). (8.31)

Classification theorem (8.32) By applying the above theory, in particular (8.29), [J. C. Wood 6] established the following theorem. Let M be a closed Riemann surface. There is a bijection between harmonic maps 56: M -+ G,(C") of finite a-order and subsets of subbundles 0 = fl, ... , $ of M x C" such that E,o rank #, = r, where fl, is a holomorphic subbundle of Ker AG.(8 0>l,

fit is a holomorphic subbundle of

and so on. (8.33) From that [J. C. Wood 6] went on to parametrise the harmonic maps M -+ Grass (C") of finite a-order by certain finite sequences U)o,,,, of holomorphic maps ft : M -+ G,,(C`i), where

t1=n,s,>0,andt1=n-2Es1>, 0 (1
The constructions are given by an explicit algorithm. They are algebraic, with

differentiation for r = 1, 2; for r >, 3 they require further the Cauchy integral transform

a--+J(' 2rci 1

a(w) dw A do

w-z

for functions a: U --+ C. (8.34) [J. C. Wood 7] gave a classification and parametrisation analogous to those in (8.32) and (8.33), with G,(C") replaced by U (n).

(8.35)

Wood]

If M is a compact Riemann surface and 0: M--+G,(C"), then [Burstall,

deg (0) = - c,(q) = - deg (0l),

the degree being that defined in (4.28).

137

453

ANOTHER REPORT ON HARMONIC MAPS

Twistorial approach (8.36) An alternative treatment of the factorisation theorem for harmonic maps S2 -+ Gr(C") was given by [Burstall, Salamon], building on methods of [Burstall, Wood]. This approach has two main ingredients. Firstly, if 0: S2 - N is a harmonic map, then 0-1Tc(N) has the BirkhoffGrothendieck decomposition (4.9) into holomorphic subbundles:

0-'T'(N) = n, LP' O+ ... O nk LPk,

where L is the Hopf bundle (c,(L) = 1) and p, > ... > pk. The integer p, -pk is the length of 0. A non-constant harmonic map has length at least 4. Secondly, as in (7.38), harmonic maps S2 -+ Gr(C") are covered by J2-holomorphic

maps into suitable twistor spaces, which are flag manifolds in the case at hand. By

exploiting the geometry of these flag manifolds and further application of the Birkhoff-Grothendieck theorem, a sequence of harmonic maps is produced via (8.24) with strictly decreasing length and thus the factorisation theorem is obtained. A key

technique in this approach is to encode the geometry of the flag manifolds into directed graphs (or tournaments). Maps into projective spaces This is a reformulation of the theorem of [Eells, Wood 4] (with preliminary contributions by [Din, Zakrzewski 1, 2], [Glaser, Stora] and P. Burns]). Take r = I in (8.32). There is a bijection between harmonic maps 0: M -+ P,_1(C) of 0-order k and holomorphic maps f: M-+ P,,-,(C) of 3-order >, k, given by (8.37)

f = G`-''(0), 0 = G(r,(f). The map 0 has finite a-order if and only if 0 is complex isotropic in the sense of is a full holomorphic map (in the sense of (7.46)) and (7.24). If f: M -+ f : M -+ G,,,(C") its jth associated curve (defined in (7.46)), then for fixed k with defined by 0 < k n-1, the map 0: M -+ (8.38)

0(x) =fk-jx) flfk(x)

is a full complex isotropic harmonic map.

is totally isotropic if the (8.39) Say that a holomorphic map f: M associated curves f, and f are orthogonal for all i, j > 0 with i+j < n- 1. If n is even, then an easy calculation shows that there are no full totally isotropic holomorphic maps f: M --+

Regarding P,-,(R) as the space of real points of

we recover the

classification theorem of [Calabi 1, 2]-which was the motivation for (8.37) and the whole twistor approach to the construction of harmonic maps of surfaces. (8.40) A map 0: M -+ P,s_,(R) is isotropic if the composition i o 0 : Mis complex isotropic. There is a bijective correspondence between full isotropic harmonic and full totally isotropic holomorphic maps f: M -+ P,a-,(C). In maps 0: M --+ particular, such maps exist only for n odd. The correspondence is given as in (8.37).

[Borchers, Garber] gave an iterative scheme for finding all such totally isotropic holomorphic maps.

138

454

J. EELLS AND L. LEMAIRE

(8.41) EXAMPLES.

Any harmonic map ¢: S2 --+ Pi_1(C) is complex isotropic [Din,

Zakrzewski 1, 2], [Glaser, Stora]. For n 3 3 there are harmonic non ±holomorphic maps S'- P._1(C) of all degrees. For n = 3 these maps are necessarily full [Eells, Wood 4]. [Bando, Ohnita] as well as [Bolton, Jensen, Rigoli, Woodward] give explicit formulas for the full harmonic maps 0: S2 --+ Prz_1(C) with constant curvature. (8.42) EXAMPLES. Any harmonic map ¢: T2 -+ Pi_1(C) of deg 0 0 is complex isotropic. For n > 3 there are harmonic non ±holomorphic maps 0: T2 -+ Pi_1(C) of all degrees [Eells, Wood 4]-in striking contrast to the case n = 2 [Report (11.7)]. (8.43) EXAMPLES. Let M be a closed Riemann surface of genus M = p. There are harmonic non-holomorphic maps M -+ Pn_1(C) of all degrees > p + 1, provided n > 3 [Eells, Wood 4].

(8.44) [Verdier 1, 2] has shown that (as a subspace of C`°(S2, S4)), the space .*'/2 k(S2, S4) = {q e.*a(S2, S4) : E(O) = 8kir) is connected. He has used the twistor

construction to endow JG ak(S2, S4) with the structure of a complex algebraic variety. Coulomb gauge fields (8.45) Let (N, h) be a Riemannian homogeneous space with G as transitive group of isometries. For each v E L(G), let v be its associated vector field on N, and define the moment map m: T(N) -+ L*(G) by m(Y) v = - . (8.46) As a consequence of Noether's theorem, [Pluzhnikov 2] and [Rawnsley 1] have shown that if (M, g) is a Riemannian manifold and (N, h) a homogeneous space, then a map 0: (M, g) --+ (N, h) is harmonic if and only if c*m is a coclosed L*(G)-valued 1 form on M; that is, d*(q5*m) = 0. (8.47)

Consider now a group G as a homogeneous space, G acting on itself by

left translation. Then the moment map m can be identified with minus the Maurer-Cartan form p (8.3). If 0: (M, g) -+ G is a map from any manifold to G, then the pull-back a = ¢*p is an L(G)-valued 1-form on M, a connection with curvature (8.48)

da+2[a Aa] = 0.

Conversely, let a be an L(G)-valued 1-form on M satisfying (8.48). It can be viewed as a flat connection 1-form on the product bundle M x G -+ M. (8.49) Suppose that Hom (n1(M), G) = H1(M, G) = 0; and fix a base point a e M. The correspondence 0 - 0*µ is a bijection between based maps 0: (M, a) --+ (G, e) and L(G)-valued 1 -forms a satisfying (8.48) [Singer].

(8.50) From the viewpoint of gauge theory, we are given a trivial G-bundle over M and a connection d+ a; and ask when it is gauge equivalent to a connection d+ a with d *a = 0. A gauge change is a map 0: (M, g) -- G; with respect to it & changes to

139

ANOTHER REPORT ON HARMONIC MAPS

455

0-ldq+0- . We observe that d*a = 0 is equivalent to the Euler-Lagrange equation of the functional J(O) =

I(

10-1dd+0-1«0I2dx.

JM

The case G = U (l) gives rise to a linear problem corresponding to equations of Maxwell fields. Those a arising from ¢ are the Coulomb gauge fields.

Totally geodesic maps (8.51)

The description of harmonic maps in terms of holomorphic constructions

described above applies only to maps of surfaces. When dim M >, 3, only special results are known, in particular in the case of totally geodesic maps (2.9). (8.52) Let N = G/K be a Riemannian symmetric space and yo e N; we have a canonical decomposition L(G) = L(K) Q T, (N) of the Lie algebra of G. If M is a totally geodesic submanifold of N (necessarily symmetric, with the induced metric) containing yo, then T"'(M) is a Lie triple system (that is, for any X, Y, Ze 7, (M) we have [X, [Y, Z]] E T,,(M)). Conversely, any Lie triple system V of T,,(N) defines a

totally geodesic submanifold M of N containing yo, and V = T;o(M); [Helgason] and [Kobayashi, Nomizu]. (8.53) The totally geodesic submanifolds of a symmetric space have been classified [Chen, Nagano 1, 2]. For instance, the maximal totally geodesic

submanifolds (a) of Sn are the equators Sn-1; (b) of Pn(C) are P._1(C) and Pn(R) ; (c) of Pn(H) are Pn_1(H) and Pn(C); (d) of the Cayley plane are P2(fl-l) and S8; (e) of the quadric Q. = SO (m+2)/SO (2) x SO (m) for m > 1 are SP x S° where p + q = m, and Pr(C) if m = 2r. For Grassmannians we have the canonical totally geodesic inclusions

Gk(Hn) -* G2k(C2n) (8.54)

`

G4k(R4n).

Any compact Lie group G can be totally geodesically embedded in a

complex Grassmannian (Rawnsley); in fact, if V is a faithful representation of G, then G --+Grass(V(D V). (8.55) It has been shown by Cahen, Gutt and Rawnsley that the complex quadric Spin (n + 2)/Spin (n) Spin (2) has a totally geodesic and holomorphic embedding in a complex Grassmannian.

Maps induced by homomorphisms (8.56)

Let 0: G -* G' be a homomorphism between compact semi-simple Lie

groups with their Killing form metrics. Then 0 is harmonic [Eells, Sampson]. Let H c G, H' c G' be subgroups, and suppose that 0: H -+ H'. Denote by

140

456

J. EELLS AND L. LEMATRE

0: G/H- G'/H' the induced map of the homogeneous spaces with their induced Riemannian structures. Its second fundamental form is given by 2V(do) (X, Y) =

where the subscript indicates the component in the decomposition L(G') _ L(H') + m'. Then [Guest 1, 21: The map 0 is harmonic if and only if z'=1 [O(ei)L >, orthonormal base of L(G). (8.57) EXAMPLE.

0, relative to an

Suppose that G is simple. Given a homomorphism

0:G-->SU(n+1), for each [v] e PP(C) define the map c,,: G --+ Pn(C) by 0v(g) = [0(g) v]; its image is the

orbit G[v] of [v]. The map 0. has constant energy density 1

m

n

e(On) _ - E Y_ I <e(ei) v, vj)12> 21-11-1

where (v5)1,1,n is an orthonormal base of [v]1. Define the function f: Pn(C) -+ R by f([v]) = e(¢); thus

f([v]) = - 2

i'-l E 0 (e,)Pv

0(e,) v, v ),

where p,, denotes the orthogonal projection of Cn+' onto [v]. Then [Guest 2] shows:

The map 0v: G -+ PP(C) is harmonic if and only if [v] is a critical point of f: P,(C) -+ R, or equivalently if v is an eigenvector of the transformation m

v -+ 0(e) (pv -P9)

t-

(8.58) Define a map v: Pn(C) -+ L*G by v = 0* o (i/2n) p, where p[v] = p,,. We call v a moment map of the G-action. Then m

2./([v])+4ir2Ip([v])I2 = -

<0(e,)2v,v>;

here F 0(e,)2 is the Casimir operator of 0. Furthermore, if 0 is irreducible, then 0,,: G --+ PP(C) is harmonic if and only if [v] is a critical point of the function lp12. (8.59) As part of a general theory of symplectic group actions, [Kirwan] has shown that 1,u12 is an equivariantly perfect Morse-Bott function-implying in particular relations between the critical indices of Jp12 and the G-equivariant

cohomology of PP(C).

[Guest 1, 2, 3] gave several variants of (8.57) as follows. Let M = G1 T, where G is a compact simple Lie group and T a maximal torus, and rk = SU (n)/S (U (r) x ... x U (rk)) where r1 +... + rk = n. Both are flag manifolds in the sense of (7.35). Call .y = . 1 ..1 the manifold of full flags in Cn, and .fir rk where o = {r1, ... , rk}. denote by 3C the natural projection (8.60)

let r

Denote by p, the orthogonal projection of Cn onto the ith subspace associated to the standard flag of .mar rk, that is, the ith space in the decomposition Cn=ChIX... XCrA.

141

ANOTHER REPORT ON HARMONIC MAPS

(8.61)

457

Let 0: G -+ U (n) be a homomorphism (that is, a unitary representation of

the induced homomorphism on the Lie algebras. Let fo: G/T-+.F be the map induced by 0 and

G on Cn) such that B(T) c S(U(1) x... x U(1)); denote by 0 Put f, = 7to of,.

(8.62) Decompose L(G/T)c = E,,, Ea, where A c L(T) is the root system and E,, is the root space corresponding to a. L(G/T) can be written L(G/T) = E,c", Vx, where A is a system of positive roots (for a given choice of a Weyl chamber), and V c = E,, E) E. On L(G/T), consider the scalar product

<x, y)r = r(a) <x, y>, for x, y e V.,

where < , > is minus the Killing form and r: A+ -* IR (> 0) a fixed function. This

product induces on G/T a left invariant metric, Hermitian with respect to the canonical invariant complex structure. (8.63) [Guest 1, 3]. Let G/ T carry the metric < , >, and 'r,_.,k the standard metric. The map fa: G/T--+JFr1 , k defined in (8.61) is harmonic if and only if the operators

E Pt' 0,,(e,) . (pt -p) ' 8*(e-a) ' P>l r(a) aeA

on Cn are all zero for I
The maps 0, are Gauss maps, associating to a point the span of the orbit of a weight vector under a set of roots. In particular, for G = SU (2) and.F, , = P,_1(C), this provides a representation theoretic description of the classification theorem (8.40). 9. Maps into loop spaces Ki hler structure on QG (9.1) In this Section G denotes a compact Lie group with bi-invariant metric. Let AG = .;(S1; G) and S1G = 2' (S',1; G, e). These are Banach Lie groups (modelled on a real Hilbert space), with Banach Lie algebras AL(G), SL(G) defined pointwise. If p, is evaluation p,(y) = y(t), p, is an epimorphism. The sequence

e

QGt AG-->G-->e Pi

is exact. Define ?7(y) (t) = y-1(1) y(t); q is not a homomorphism, but G = 17-1(e). Note

that rl o i = Idn,. Thus we obtain the identification (9.2)

SIG = (AG)/G;

and AG is the semi-direct product S2G>< G. (9.3) The Maurer--Cartan form Pn, on SIG is a 1-form with S2L(G)-values. Indeed, consider the restriction p,: Q G -+ G of the evaluation map; then t -, p* µ. is

142

J. FELLS AND L. LEMAIRE

458

a loop of 1-forms on SfG with values in L(G), and therefore defines a 1-form p*/cc on SZG with values in S2L(G); and (9.4)

Pnc = P*Pc

(9.5) Define the left invariant 2-form S on S2G, given on left invariant vector fields g, ?j by

S(,tl)=fs The form S is nondegenerate and closed : dS = 0. Otherwise said, S defines a left invariant symplectic structure on SIG.

Viewing the Maurer-Cartan form pnc as a map of S' into the vector space of 1-forms on SiG with values in L(G), we can take its tangent map tinc, and write S in the form 1

S= 2 f,,A (9.6)

Let G' be the complexification of G. Put 9 = {y e AGc having a

holomorphic extension over the closed unit disc D c Q. Then (AG)c/9 is a complex homogeneous space homeomorphic to SZG [Pressley 1], [Atiyah] :

92G = (AG)/G

(AG)c/Y

II

II

AG (AG) n 9 (9.7)

(AG) Y

9

Elements e IL(G) can be represented by Fourier series

= E. nmo ba X.,

with cn E L(Gc) satisfying the reality condition _n = cn and Xn = e-2ntnt -1. (9.8)

Define the operator J on fL(G) by

Ea

in

bnx ;

and extend it by left translation over SIG; then J is multiplication by i on + frequencies ; and by - i on - frequencies. (9.9)

The operator J is a left-invariant integrable complex structure on SiG. The

form k(c,)i) = on 92L(G) is symmetric and positive definite. It gives a smaller topology than that of 2;; in fact, k determines the Y' ,,metric on fL(G). Nonetheless, we shall speak of (SIG, k, J) as the Kahler structure of SIG [Pressley 1], [Atiyah].

SIG as a homogeneous space [Freed 1, 2] (9.10) Let S' act on AG by rotation of the parameter. The semi-direct product S'
e---+ S' x G--,. S1

AG-r--+fG

e,

143

ANOTHER REPORT ON HARMONIC MAPS

459

where the homomorphism r(eie, y) = y(1)-'y and we have identified G with the point loops in AG. Now S' x G is the centraliser of a circle, so (9.11)

(S'>< AG)/(S' x G)

represents S2G as a flag manifold of S'AG. The symplectic form S is invariant under

the action of S'>< AG. (9.12)

The Kahler structure on QG can now be described in terms of the

complexification (L(S') >< AL(G))c = (O Q L(G))c p m+ e m_, where the ± holomorphic parts m+ consist of the loops on Lc(G) with strictly T- Fourier expansions :

m± _ E Lc(G)'.

0.

n>O

(9.13)

The global action of S' i< AG on f2G has infinitesimal form which assigns a vector field Z to each Z e R >< AL(G); evaluation at eefIG identifies m+ a m_ with Te (S2G). We obtain a map 0: (R >< AL(G))C ---+ End m+

as the C-linear extension of (9.14)

X

(V

m+,

V denoting the Kahler connection and 2 the Lie derivative. (9.15)

For any map Z: S' -* Lc(G) the Toeplitz operator TZ: m,, -+ m+ is defined

by

(9.16)

TZ(Y) = [Z, Y]+,

the subscript now denoting projection onto the strictly positive components of the Fourier series. [Freed 1] has expressed the connection and curvature of the Kahler manifold (S)G, k, J) in terms of Toeplitz operators, through the following formulas and (9.16) : 8(H) = TN for He (18 Q L(G))c,

6(X)=TxforXem_; 9(X)=- TX forXem+. The curvature is the End(m+)-valued (1, 1)-form R(X, Y) = 0([X, YD - [O(X), OVA,

Freed has shown that R is (in a delicate sense) traceable, and is therefore able to define

the associated Ricci curvature form. (9.17)

Freed also uses these Toeplitz operators to produce an explicit complex

Fredholm structure on S2G (in fact, a reduction to the group GL' of units in the algebra

of trace class operators on QLC(G)). From general principles [Eells, Elworthy] we have Chern classes of SIG as a Fredholm manifold. And it is shown in [Freed 1] that its first Chern class cl(S2G) is expressible in terms of its Ricci form.

144

460

J. EELLS AND L. LEMAIRE

The energy function

(9.18)

The energy of a loop y E AG is E(y) =

zI

IYI2 dt,

s

with associated Euler-Lagrange equation d (y-'y) = 0; equivalently, y(t) = y(O) exp (at)

for some a e L(G). Its solutions are the closed geodesics of G, parametrised proportionally to arc length. (9.19) Restrict attention to E: Q G -> R. For any E-critical point y : S' -+ G, we evaluate the Hessian V2E(y) on u, v E S2L(G) : VIEW (u, v) = S(u, v + [y-'7 , v]).

This gives the Lax form

v+[y-'y, v] = 0

(9.20)

of Jacobi's equation for y. The Hamiltonian vector field XE associated to E and S is characterised by dE = XE j S. Its Hamiltonian flow is given by rotation of loops: s - Q8(y) t = y(t+s) y(s)-'. Its fixed points (the critical points of E) are identified with the homomorphisms S' -> G. (9.21)

(9.22) The critical set Crit E of E is a disjoint union of closed submanifolds, and on each of these the Hessian V2E is nondegenerate in normal directions. Also, VE = -JXE is the gradient of E with respect to the Kahler structure (9.9). Its trajectories determine a decomposition of S2G with very rich structure [Pressley 1,

2], [Pressley, Segal]. (9.23)

Every conjugacy class C of the adjoint representation of G on L(G) has

the form C = G/H` where H` is the centraliser of a torus (= the closure of the ]parameter subgroup generated by an element c e C). The map F: C S2G given by F(c) = exp tc is holomorphic and totally geodesic. Each component of Crit E is a conjugacy class C in SiG as a complex submanifold. Its invariant complex structure is induced from the Kdhler structure of f1G.

CIG as a twistor space (9.24) From the Fourier expansion (9.7) of E b2L(G) we see that the complex structure J = J, is represented in the form J1

d

= Y- is xn+ n>O

Thez reality condition ensures that ` 1((S n)n>0) = (i )n>0

Xnn
i;

is determined by (in)n>o and thus J, by

145

ANOTHER REPORT ON HARMONIC MAPS

461

There is another left invariant almost complex structure J2 on QG which is important to us. For e CIL(G) it is given by (9.25)

(-1)rt+liYnxn+

J2S = n>O

(-1)nl rzXn' n
S

Thus J2 acts as J, on odd subscripts, and as -J, on even subscripts. As in (7.5), J2 is never integrable. (9.26) Consider the evaluation map p = p-,: SAG --> G. This is a locally trivial smooth fibration; and it is equivariant with respect to G-action (conjugation) on CIG and on G. Burstall has found : (9.27) The map p: (C2G, J2) -* G is a twistorfibration. In particular, if u: M -+ QG is a J2-holomorphic map of a Riemann surface M, then 0: p o yr : M -+ G is harmonic.

The twistor fibration in (9.27) has the following universal property. If K` denotes the centraliser of exp nc and H` is defined as in (9.23), then G/K` is a Riemannian symmetric space, and H' c K`. The natural homogeneous fibration n: G/H` -+ G/K` is a twistorfibration with respect to J2, obtained as in (7.35) [Burstall, (9.28)

Rawnsley 3].

(9.29) Now G/K` is naturally identified with the conjugacy class of exp nc in G. Furthermore, with F as in (9.23), we have the commutative diagram

G/II°

F

--b- !QG

G/K°

. G,

and F is both J,- and J2 holomorphic. Thus, the restriction of 0 to G/H° is the twistor fibration of (7.37). Families of maps (9.30) As in (8.3), let 0: M -+ G be a map of a Riemann surface. For A E C* we form the LC(G)-valued 1-form on M:

a, =

2(1-.Z 1)a'+2(1-2)a

.

(9.31) If 0: M -> G is a harmonic map, then a,, satisfies the structural equation dal+2[a2 A a,] = O for all AEC*. Otherwise said, or, is a flat G-connection form on M. As in (8.10) we have [Zakharov, Shabat], [Uhlenbeck 5] : (9.32) Let M be a Riemann surface with Hom (n,(M), G) = 0; fix a base point a e M, and let 0: (M, a)--+ (G, e) be a harmonic map. Then there is a unique map

t : C* x M -* GI satisfying the linear system (9.33)

d(D, = d> a, (equivalently, ( D - 1

,. =

with 4), = e, b_, = 0,',i(a) = e for all AaC*. Furthermore, CD is holomorphic in A.

Indeed, (9.31) is the integrability condition for (9.33).

146

462 (9.34)

J. EELLS AND L. LEMAIRE

Consider the restriction of 2 to S' c C ; write A. = e2rz", and

'z'dI = (1-1') I

18'IA+2(1-2)c1

Then I 18'(D, = - zXi az, so the restriction (D I S' x M --> Gc defines a map (D: M --> SIG called the canonical lift of 0: (SIG, e)

zP

(M, a)

. (G, e).

Since a'c = 0(- k, a'), we obtain : (9.35)

The canonical lift cD is both J,- and JZ holomorphic.

(9.36) Items (9.30)-(9.35) can be used to summarise [Uhlenbeck 5]'s construction. With every harmonic map 0: S2 -- G we associate its canonical lift

0: S2 - SIG, which is J,- and J2-holomorphic and satisfies (9.37)

Im ((D-'d(D) c Lc(G) ® Span (A. `- 1, A - 1).

Conversely, any such map cb defines the harmonic map 0 = D(-1) : S'-+ G. [Valli 2] has interpreted the canonical lift as a geodesic in the gauge group '(M, G), where G = U (n). (9.38) Let G be a compact simple group. Then H2(S G; 7L) = Z. Its positive generator is a multiple of the class of the Kahler form co'°. A direct calculation by Eells, [Freed 1] and [Valli 1] shows: (9.39) energy

If 0: S2 -> G is harmonic and 'D: S2 -, SIG its canonical lift, then the E(95) =

67 deg 0, 101,

where 1012 is the length' of the highest root of G. (9.40) Define c: S2 x S1 -+ G by q(x, t) = p,((D(x)) and put 6 = . Then if 0: M -> G is harmonic, 3E(c) = fs,xs' cS*6 [Hitchin 2]. Hitchin also obtains a formula for the energy of a harmonic map 95: T2 --> SU (2) in terms of its spectral curve. (9.41) Let G/K be a Riemannian symmetric space of compact type, and a the involution of G whose fixed point set is K. Then the map is G/K- G defined by

i(gK) = a(g) g ' is a totally geodesic immersion. Define the subgroup

A(G,a) = {yeAG: a(y(z)) = y(-z) for all zeS'}.

147

ANOTHER REPORT ON HARMONIC MAPS

463

From (9.2) we have the commutative diagram

A(G, v)/K -! AG/G = QG lp

G/K-`

-

G,

where j(yK) = y(y(l))-' and p(yK) = y(1) K.

Then j is an embedding which endows A(G, a)/K with a Kahler manifold structure. [Rawnsley 3] proved that p is a twistor fibration. The canonical lift
S' action and Jacobi fields (9.42) The circle group S' acts on SIG by rotation of the parameter, in the following sense: (e{8y) t = y(t + s) y -1(s) for each y e G. That action induces an S-

action on maps
(etsb)-1d(e{8D) = C e2'' AdI(s)(SnQ J noo

Xn

Thus the S'-action preserves Jl- and JZ holomorphicity. (9.43) Let b : M -+ SIG be the canonical lift of 0: M --+ G and suppose that S' - (D = (D. Then (1) t(M) is contained in some flag manifold G/H; (2) c(M) is contained in some inner symmetric space G/K c G (inner means that

rank G = rank K). (9.44)

The Hessian of a harmonic map 0: M G has the form H0(v, w) =

(< dv,

dw) + ([*pv], w>)

,

where p is the Maurer-Cartan form of G. If Hom (trl(M), G) = 0 we can write *¢-'do = dot,

*dq¢-' = dfl

for suitable maps a, f : M -+ L(G); and we can choose a(a) = 0 = fl(a) if 0(a) = e, 4),(a) = e, in which case a,fl are unique.

With the indicated notation for left and right translation in G we have the following result of Rawnsley: (9.45)

x -+ XX(x) =

a(x) + (R,(Z))* fl(x)}

is a Jacobi field along 0: M -+ G. The assignment 0 --+ XX defines the infinitesimal generator of the S'-action on the space of harmonic maps M -+ G.

148

464

J. EELLS AND L. LEMAIRE

Instantons (9.46)

There are close relationships between G-instantons and holomorphic

maps S2 --> S2G [Atiyah], [Donaldson 1]. An exposition of these and related ideas is given in [Donaldson 2]. Very briefly, we can: (1)

identify based gauge equivalence classes of G-instantons on ll = C2 with

based isomorphisms classes of holomorphic G'-bundles on P2(C) which are trivialised on a projective line P1(C); (2) identify those (using their transition maps) with the based holomorphic maps S' -+ 92G.

10. Maps into spheres The phenomena encountered in (2.5) show that energy minimising techniques do

not apply to harmonic maps into spheres of dimension > 3. However, there is an existence theory exploiting the special features of the harmonic map equation (10.2) below. (10.1)

For any map 0: (M, g) - S', denote by 1 its composition with the

standard inclusion i: Sn - Rn+i : (M, g)

0

- Sn

li

Rn+I

Then 0 is harmonic if and only if (10.2)

t1(D = Id(1)12(D.

In the presence of sufficient symmetry (10.2) can be reduced to an ordinary differential

equation. Most of this Section examines harmonic maps in that context. (10.3)

Say that a harmonic map 0: (M, g) -S' is an eigenmap if it has constant

energy density. Polynomial maps (10.4) Consider a map im -+ Rn whose components are harmonic polynomials, homogeneous of the same degree k, and suppose further that it maps S' into Sn'Then

its restriction (D: S'"-' - In satisfies 0M = ,1I, with A = k(k+m-2), and thus induces an eigenmap 0: S'"-' -,. Sn'In fact, the description of all eigenfunctions of the Laplacian on S"`-' implies that any eigenmap 0: Sm-1- Sn-' is of that form [Report (8.1)]. (10.5) EXAMPLE. An orthogonal multiplication f: RP x R" -+ In is a bilinear map satisfying If(x,y)I = Ixl'IYI [Report (4.16)].

If p = q = n and n = 1, 2, 4 or 8,.then f is a harmonic morphism (2.30); and conversely [Baird 1].

If p = q, then the quadratic map 0: S271-' -* S", defined via the Hopf construction O(x, y) _ (Ixl2 - I yI2, 2f (x, y)) is a harmonic polynomial map [Report (8.4)]. Also, 0 is a

149

465

ANOTHER REPORT ON HARMONIC MAPS

harmonic morphism if and only if p = q = n = 1, 2, 4 or 8 [Gigante]. In fact, up to isometries, the Hopf fibrations are the only quadratic harmonic morphisms between spheres [Yiu]. (10.6) [Parker] has classified the orthogonal multiplications V x R" -> 68n for p=2,3. [Toth 4] has parametrised the range-equivalence classes of such IJ

full orthogonal multiplications (p < n <, p2) by a compact convex body LP in a finite dimensional SO (p) x SO (p)-module

VD,

identified

via Ad Q Ad with

L(SO (p)) x L(SO (p)). We have

dim L'/(SO (P) X so (A = P2(4 1)2-P(P-1) forp>, 3. (10.7)

An orthogonal multiplication of the form J1

x 68n -+ 68" is associated

with a Clifford module of the standard Clifford algebra of W". It determines a totally geodesic embedding S' -- O (n). [Q-M. Wang 2] has shown that every homotopy class

in np(O(oo)) has such a totally geodesic embedded representative of constant curvature. Here 0 (oo) = limn,, O (n), following the standard inclusion 68" --> p8n+1 (10.8) Recall [Report (8.5)] that a function f: Sin-1 -+ 68 is isoparametric if IV fl2 and Af are both functions off. If to R is a regular value, then f-1(t) is a hypersurface of constant mean curvature, called an isoparametric hypersurface. (10.9) An example of an isoparametric function is the restriction to Sm-1 of a harmonic k-homogeneous polynomial P: Rm -. 68 with the eiconal property IVP(x)12 = Ixl2k-2 for all xE Rm. Clearly, VPISm = : Sm-' - Sm-1 is a harmonic

polynomial map. (10.10) EXAMPLE [Baird 1]. Let 0: Rm -- 68" be a k-homogeneous polynomial map which is a harmonic morphism. Normalise 0 so that sup{1gf(x)I2: xeSm-1} = 1. Then m - 2 > (n - 2) k, with equality if and only if 1012 =I on Sm-1. The --> R defined by f(x) = 10(x)12 is isoparametric. The level function f: M = {x E Sm-1: Ic(x)I2 = 11 is a smooth (not necessarily connected) minimal submanifold of Sm-1. The map 0: M -+ Sn-1 is a harmonic Riemannian submersion. On the cone over M with vertex 0, we have the eiconal property 22(x) = p2lxl2k-2, where ,12 is defined by
(10.11) ExAMPLE. Let P0, ... , P. be symmetric endomorphisms of IR with P P + P P, = 26,i 1 (0 < i, j <, m). That is called a Clifford system on 6824. Each

representation of the Clifford algebra of (l m-1, - ( , )) on R4 is characterised by skewsymmetric endomorphisms E1... Em_1 of R4 with

E,Et+E1E,=-26,11(1 <, i,j<m-1). These define a Clifford system by setting P,(u, v) = (u, - v), Pi(u, v) = (v, u), P,+t(u, v) = (E, v, - E, u). All Clifford systems are so determined. Set f,(x) =

. Then [Ferus, Karcher, Miinzner] the function f: 6824 - 6B defined by f(x) = x14-2 E; 0 fl(x) is isoparametric.

150

466

J. EELLS AND L. LEMAIRE

(10.12)

If Sm denotes the unit sphere of the vector subspace of End(R211)

spanned by P0, ..., Pm, then we have the totally geodesic embedding Sm -> Gq(R22), assigning to each P E Sm its + I eigenspace in Req. Defining the infinite Grassmannian BO (oo) = lima-00 Gq(R27) = classifying space for the infinite orthogonal group 0 (cc), every homotopy class in tcm(BO (co)) has such a totally geodesic embedded representative of constant curvature. [Q.-M. Wang 2]. (10.13)

Let M be a compact hypersurface in S"+' and let V(S "+', M) be its

normal bundle, as a subbundle of T(S"+') 1119. Denote by rl: V(S"+', M) -+ S"+' the restriction of the exponential map exp: T(S"+') -+ S"+' Its critical points (= critical normals) form a set C,, c V(S"+',M). Call M totally focal if q-'(rl(C,)) = C,,. Results of [Cecil, Ryan] combine with those of [Carter, West] to show that a hypersurface M in S"+' is isoparametric if and only if it is totally focal. (10.14) Let f: (M,g) - R bea smooth function satisfying Idfl2 = Aof for some function A : R -+ R. Then [Q.-M. Wang 1]

(a) (b)

the focal varieties off are smooth submanifolds of M; each regular level set off is a tube over either of the focal varieties;

(c)

in case (Mg) is either Sm or Rm (but definitely not RHm), f is in fact

isoparametric. In that case the focal varieties are minimal submanifolds. (10.15) EXAMPLE.

Let f: Cm - C be a k-homogeneous polynomial having 0 as an

isolated critical point. Then M = f-'(0) n S2m-1 is a minimal submanifold of real codimension 2 in S2m-'. For instance, take f: C3 - C of the form f(z) = z; +z2 +z3 (k > 2). Then M =f-'(O) n S5 is a Brieskorn 3-manifold, minimally embedded in S5 and stable under the canonical S'-action on S5. The quotient M/S' = N is a

Riemann surface of genus (k-1)(k-2)/2, and the quotient map 0: M-*N is a harmonic morphism (Baird).

A reduction theorem (10.16)

Let p: (M, g) - I be an isoparametric function such that on M*

M\{xeM: Vp(x) = 0} (i) the integral curves of the normal vector field c = Vp/lVpl are geodesics; (ii) the principal curvatures of each level surface M, = p-'(s) are constant; (iii) the differential of the projection M, --- M8 along integral curves of c maps the principal spaces of M, (that is, the eigenspaces of its second fundamental form /,) to those of M,..

If a: (N, h) -+ J is another such isoparametric function, we shall say that a map 0: (M, g) --> (N, h) is (p, a)-equivariant if there is a function a: I -+ J such that

a o p = or o 0; and if do maps normals to normals.

(10.17) We have a reduction theorem of [Baird 1], [Pluzhnikov 1], in a form given by [Karcher, Wood]. Let 0: (M,g) -+ (N, h) be a (p, a)-equivariant map. Then is harmonic if and only if (a) each map 0, = 01 M, --+ N,,(8) is harmonic ; (b) on M*, at satisfies (10.18)

a"(s) - (Trace,, f,) a'(s) 4 Trace

do,) = 0.

151

ANOTHER REPORT ON HARMONIC MAPS

467

Note that (10.18) can be written in the form (p(s) a'(s))' + w(a, s) p(s) = 0,

where p(u) = exp fu, -Op. That is the Euler-Lagrange equation of the functional F(a) = J(p(s) a'(s)2 - W(a, s)) ds, where W(a, s) = f( w(c, s) p(s) dd.

If each level NN is a Euclidean sphere, then the third term in (10.18) is a multiple of e(0,), which is constant on M, if 0 is harmonic. Several technical obstacles remain before effective application of the reduction theorem can be made. Here are some favourable instances. (10.19) EXAMPLE [Smith 1].

The join of 2 Euclidean spheres

Sm-1 = SP-1

* Sr-1 (p+r = m)

is obtained by representing the points z e S"`-1 in the form z = (sin s x, cos s -y), with x e SP-', y e Sr-1 and 0 < s <, n/2. The function p: S in 1- 11 given by (sin

is isoparametric; in fact; IdpI2 = 1,

-Ap = (r-1) tans-(p-1) cots.

Its level hypersurfaces are given by M. = sins S"-1 x cos s Sr-1 (0 < s < 7Z/2). The join of two maps u: SP-1 -+ Sq-1,

v: Sr-1

SQ-1

is defined by 00 = u * v = Sin-1-+ Ss-1, where 0o(z) _ (sin s - u(x), cos s v(y)). (10.20) Suppose now that u, v are eigenmaps as in (10.4), with eigenvalues A. and A.. Then with an isoparametric function a: Si-1 --+ R analogous to p we see that 0o is (p, a)-equivariant, with an defined by ;(s) = s (0 < s <, n/2). We take

M* = S'"-1

-(Sp_1

x 0) U (O x Sr-1),

N* = Si-1 _ (S11-1 x 0) U (0 x Ss-1)

in (10.16). Now a change of parameter a: I -- J produces a new map 0: M* -+ N* by 0(z) = (sin a(s) . u(x), cos a(s) v(y)), which is also (p, a)-equivariant. We have r(0) = a"(s) + (hp) a'(s) + k(s) sin 2a(s), where

k(s) = 2 (c

s2 s

sine s)

We note especially that k(s) sin 2a(s) is independent of x e M,. And that 0,, is harmonic because u, v are. [R. T. Smith 1] gave conditions under which the equation (10.21)

a' (s) + (Op) a'(s) + k(s) sin 2a(s) = 0

can be solved for a with a(O) = 0, a(n/2) = rz/2; then the associated map 0: S'"-1- Si-1 is harmonic and homotopic to 00 (see [Report (8.7)] or (10.25) below). Writing (10.21) in the form Au(s) - k(s) sin 2a(s) = 0 displays the analogy with the sine-Gordon equation.

152

468

J. EELLS AND L. LEMAIRE

A similar construction for maps 0: Sm-' -> H"-' leads to an analogue of the sinh-Gordon equation. (10.22) EXAMPLE [Baird 1].

The equivariant maps of ER"` to S'° studied in (12.25)

below can also be interpreted in this framework by writing 0: Et --r S' in the form

0(s-x) = (sin a(s),cosa(s)-x) for xeS"t", se R(? 0) and a(0)= 0. (10.23)

The following simple construction is due to Baird and Eells.

Let u: (M,g) -+S°-' and v: (M,g) -> S8-' be eigenmaps, corresponding to the eigenvalues .lu and av of A on (M, g). If 0: (M x S',g+ds2) --> S°+8-' is defined by ¢(x, s) = sin a(s) - u(x) + cos a(s) v(x),

where a is a suitable solution of the pendulum equation sin a cos a = 0, a- ('L then is harmonic. Indeed, treating 0, u and v as maps 0, U and V into RQ+8, R9 and R', we have A(D =

0e,2

where 0 = cos a U- sin a V. Then 1012 = 1 and <(D, 0> = 0. Since A(D is in the plane

spanned by t and 0, we see that 0 is harmonic if and only if 0 = = (Au - A,,) sin a cos a - o(. EXAMPLE. If u, v are minimal Riemannian immersions of (M, g), then A14 = m = 2n. Consequently, the map 0: M x S' -+ Sp+q-' given by O(x, s) _ (sin s - u(x). cos s - v(x)) is harmonic. Other related constructions of harmonic maps St-2 x S' -+ S"-' have been found by Uhlenbeck and [Hsiang, Yu].

(10.24)

Another framework for reduction is the following. Let G be a group of

isometries of both (M, g) and (N, h); and assume M compact. Then G acts on '(M, N) with fixed point set '(M, N)G consisting of equivariant maps; and E: f(M, N) - ER is G-invariant. For any q0 e '(M, N)v, the variation ot(x) = expoowtt(q50)(x)

is in '(M, N)G for all t >, 0; and

dE(0) I

t-0

= - I) M IT(O0)I2 dx.

Consequently, every critical point of EV)c is a critical point of E. That is a special case of a theorem of R. T. Smith [Report (4.17)]-an example of symmetric criticality in the sense of [Palais]. Maps between spheres

Let u: SP-' -SQ-' and v: S'-' -> S8-' be eigenmaps with eigenvalues A. where ku,k are their polynomial degrees. We examine conditions to ensure that their join u * v: Sp+r-' -+ S9+8-' can be deformed into a harmonic map. As in the reduction theorem, we look for a solution of the form (u*av)(x, y) = (sin a(t)-u ( -),cosa(t)-v(y)) (10.25)

and .1"; thus .lu = ku(ku+p-2),2 =

IYI'

153

469

ANOTHER REPORT ON HARMONIC MAPS

for (x,y) e(l '-0) x (184-0) and t = log lxl/jy . The harmonic map equation is thereby reduced to finding a solution a: 18 -> (0, n/2) of

a(t)+(p2)ee+e(`

(10.26)

with

(10.27)

-2)e'a(t)+A"e`+e.`e `sina(t)eosa(t) = 0,

lim a(t) = 0,

lim a(t) = 7t/2

t--00

t..+o]

[R. T. Smith 1].

For 1, > 0 and ,, > 0, that occurs if and only if the damping conditions

(10.28) (a)

(p-2)2 < 42w

(b)

(r-2)'<4A,,

or 2k < (p-2)- (p-2)2-42u, or

2k.,, < (r-2)- (r-2)2-4..,,

are satisfied [Pettinati, Ratto]. (10.29) That characterisation completes the fundamental theorem of [R. T. Smith 1] (see [Report (8.7)]). Using the solution a of (10.26), two eigenmaps u: SP-1-+ SO-', v: Sr-1--> Ss-1 can be harmonically joined (to give a harmonic map u*av: SO"-1) if and only if damping conditions (10.28) are satisfied. SP+r-1

(10.30) Take v = Id,r the identity map on Sr-1. Then u* Id,, is homotopic to the rth suspension of u. Now .lv = r- 1, so its damping condition is satisfied if r < 6; therefore [R. T. Smith 1], [Ratto 3, 4], any eigenmap u: SQ-1 can be Sp-1

harmonically suspended 6 times. (10.31)

The idea of the proof of (10.27) is the following. Define oc4t : (0, n/2) -, (0, + oo)

as follows. Let to be the point where the gravity of (10.26) is 0. For any ao a (0, n/2) let 4 (oco) be such that the solution of (10.26) with initial conditions a(t0) = ao and a(to) = oca(oco) (resp. a(to) = oco(ao)) grows to 7r/2 as t oo (resp. to 0 as t -> - oo). We compare (10.26) with

a(t)+

(p-2)e `-(r-2)e` e`+e `

6:(t) +

(r-1)e`-(p-1)e ` sin a(t) cos a(t) = 0. e`+e-`

That is the equation for harmonicity of Ids,,+,. The key observation is that their gravities satisfy

.lue e'+ e`

(r-1)e'_

(p-1)e'

e'+ e-'

for all t e R. Then a standard comparison theorem ensures that there is an &e (0, n/2) for which 4(6e) = A simplified proof of (10.27) has been given by [Ding 2]; he interprets the non-

existence assertion in terms of the mountain pass theorem. For sufficiency he minimises the reduced energy over maps of the form u *Q v. (10.32) EXAMPLE.

Consider the map uk: S1 -+ S' of (topological) degree k defined

in complex variables by z -+ z' for k >, 0 and z - z -k for k < 0. By (10.30), it can be suspended 6 times to harmonic maps of the same degree. Thus [R. T. Smith 11: for m < 7, every homotopy class of maps from S' to Sm has a harmonic representative.

154

470

J. EELLS AND L. LEMAIRE

(10.33) EXAMPLE.

[Takagi, Takahashi] have obtained as gradient of an

isoparametric function a harmonic polynomial map v: S' - S' which is of polynomial degree 5 and topological degree 1. By (10.27), it can be joined to the maps uk above, so that every homotopy class of maps from S9 to S9 has a harmonic representative [Fells, Lemaire 5]. (10.34) EXAMPLE. The Hopffibrations u: S2m-' - S'n are eigenmaps (m = 1, 2, 4, 8), which can be harmonically suspended 6 times. That follows from (10.30); we remark

that the case m = 8 could not be included in Smith's version. (10.35) EXAMPLE.

If the eigenmaps u, v have the same polynomial degree (that is,

k = kv), then their join u * v: SP"-' - Sq+B-1 can be deformed to a harmonic map (Smith proved that in case p = r). (10.36)

[Ratto 41 modified his arguments in (10.30) to obtain the following

theorem. Let f: SP-' x Sq-1 -+ S"' be a harmonic map with constant energy densities 2P/2, ;q/2 in each variable separately. Then the Hopf construction on f can be deformed to a harmonic map Sp+q-1-+ Sn, provided that either of the damping conditions (10.37) is satisfied :

(10.37)

((p-2)2> 42., and (q-2)2 > 4Aq) or (p = q and AP = 2q).

(10.38) EXAMPLE. If f is the restriction of an orthogonal multiplication, then = p-1, .tq = q- 1. For p, q > 6 the Hopf construction deforms to a harmonic map SP+q-' - S". Together with a theorem of [R. Wood], that ensures that any quadratic polynomial map S'n -+ S" is homotopic to a harmonic map provided m > n + 6. (10.39) EXAMPLE.

The J-homomorphism JP : nP(O (n)) - nP+n(S ") is defined via

the following construction. For a: SP -+ O (n) let fn: S" x Sn-' -+ Sn-1 be given by fn(x, y) = a(x) y. Then denote by JP(a): SP+n -+ Sn the Hopf construction on fn. Take the limits nP = limn. nP+"(S"), JP: it (O (co)) -+ 7r,. Then JJ nP(O (n)) is stable for n -1 > p; and injective for p = 0 or 1 mod 8. Im J,, is a finite cyclic group [Adams 1]. It is known through work of Baum and Hefter that Im J1, c it,, consists precisely of those stable homotopy classes which can be represented through the Hopf construction on an orthogonal multiplication 18P+' x I{84 -+ {18", for some n. [Ratto 4] has applied (10.36) to establish that any element in ImJP can be harmonically represented, provided 6.

More precisely, if ae n,(O (oo)), then for some integer n with n- I > p > 6 there is a Clifford multiplication fn: RP" x Rn -+ Rn for which JP(a) in nP+"(Sn) has a harmonic representative. For instance, if p = 8i (i >, 1) and n = 2", then 712 = Im J,, c nP+n(S4), and by (10.36) the generator has a harmonic representative. Similarly for the other cyclic groups ImJP for p = 1, 3, 7 mod 8. From (10.36) we see that the map constructed is even, and so defines a harmonic map PP+n(R) -+

(10.40)

Sn.

See Items Added in Proof, p. 501.

155

471

ANOTHER REPORT ON HARMONIC MAPS

Rendering (10.41)

As in (5.7), say that a homotopy class of maps can be rendered harmonic

if there are metrics on both manifolds such that the homotopy class contains a harmonic map for these metrics. As an extension of (10.32), [Ratto 2] has shown that each homotopy class of maps from S' to S' can be rendered harmonic, without limitation on m. More precisely: (10.42)

Let g, h be the Euclidean metrics on Sp-1, Sr-1. We consider now

Riemannian metrics on the join Si'' -* Sr-1 =

SI"'-1 of the form

sin's where k: [0, r/2] - R(> 0) is a smooth function [0, e] U [n/2-e,n/2] for small e > 0. Let K(t) = k(arc tan e`). Now the harmonicity equation takes the form

for s E [0, 7c/2],

[(P_2) e -(r2)r'1 rl(t)+`+e`

(10.43)

e t-e` K(t)]A(t)+KZ(t)I

-1

on

sinA cosA = 0.

The idea is to choose any smooth strictly increasing function A that covers the prescribed range. Then (10.43) can be written K(t)-12PA(t)K(t)+12QA(t) K3(t) = 0.

Hence setting y(t) = )C-2(t), Y(t)+P,,(t) Y(t) = QA(t).

We look for-and find-an A so that this has a solution Y with Y(t) > 0 for all t c- 11 and Y = I if Iti > C for some C >, 0. Consequently [Ratto 2, 4] : Sp+r-1- Sa+B-1 of eigenmaps can be (10.44) The homotopy class of any join u * v: rendered harmonic. In particular, every class in it (Sm) can be rendered harmonic.

- Ss-1 is a In the same order of ideas, he proves that if f: SP-1 x map such that the restriction to each factor is an eigenmap (2 < p, q), then its Hopf (10.45)

S°"1

construction Sp+q-1-+ S" can be rendered harmonic.

(10.46) A somewhat different argument is required to show that every class in ic3(S2) can be rendered harmonic [Ratto 1]. And similarly for 7tn+1(S") (n >, 3).

Properties of maps into spheres (10.47) [Solomon]. Let 0: M-+ S" be a harmonic map of a compact manifold to S", and let Sn-2 be a great sphere of codimension 2 in S". If c(M) does not link or meet Sn-2, then 0 is constant.

The proof begins by showing that S"\S' 2 is a warped product. Then 0 lifts as a harmonic map from M to the universal cover of S"\S"-2, and a maximum principle implies that it is constant.

(10.48) [Ramanathan 2]. Let 0: S' --+ S' (m > 3) be a harmonic map not homotopic to a constant. Then E(c) = max {E(o o y) : y e G} > 2m Vol (S'"), where G is the group of orientation preserving conformal automorphisms of S'.

156

472

J. EELLS AND L. LEMAIRE

(10.49) Let V (M,g) --> R" be a non-constant map of a compact manifold; and 0 = (Do+ E'O1 b its spectral representation (relative to A', so the spectrum has the form 0=4<.1i < ... and (Di is an eigenfunction associated to A). Then (Do is the constant vector f,,, (Ddx/f,,, dx, called the centre of mass of (D. If the representation is finite, = (D' + Ei P 4f, where p is the smallest positive index with 0, * 0 and q the largest one, we say 'V has order [p, q] ; and has k-type if exactly k of the 'f are * 0 (.I%1).

(10.50) [Chen, Morvan, Nore]. 2J., E('V) _< f m 10'V12 < 2A E('V), where [p, q] is the order of 0. Either equality holds if and only if 'V has 1-type. As a consequence:

(10.51) Let 0: (M, g) -* S"-1 be a map for which 'o is the centre of S"-1. Then E(O) >

vo1(M, g)

with equality if and only if 0 has order [1,21].

Minimal immersions Mm - S" For general properties in case m >, 3 see [Chern, do Carmo, Kobayashi), [W. Y. Hsiang 1, 2], [Lawson 3], [Simons]. (10.52) Every compact Riemannian homogeneous space can be minimally immersed in some Euclidean sphere. See the references in [Report (4.14)].

(10.53)

In the next results the techniques involve

special properties of

transformation groups of isometries (especially, their principal orbits) to reduce the harmonic map system to an ordinary differential equation. For various n > 4 there are minimally embedded (n-1)-spheres in S" which are not totally geodesic. [W.-Y. Hsiang 5, 6] for small values of n, and [Tomter 1, 2] for all even n. We have seen in (8.40) that there are none for n < 3. A somewhat different method, involving isoparametric families and producing different types of examples, has been exploited by [Ferus, Karcher]. They also foliate R" by complete minimal hypersurfaces which are regular except for one absolutely minimising cone. (10.54) In related directions, there are: (1) codimension I minimal immersions [W.-Y. Hsiang, Sterling]; isotropic

minimal immersions [Tsukada]; (2) codimension 2 minimal immersions of spheres and exotic spheres in S",

with distinct knot types [W.-T. Hsiang, W.-Y. Hsiang, Sterling], [W.-T. Hsiang, W.-Y. Hsiang 4]; (3) minimal hypersurfaces in symmetric spaces [W.-T. Hsiang, W.-Y. Hsiang 1, 2], [W.-T Hsiang, W.-Y. Hsiang, Tomter] ; (4) hypersurfaces with constant mean curvature in S" [W.-Y. Hsiang 4], [W.Y. Hsiang, Teng, Yu 2]; in 111" [W.-Y. Hsiang, Yu], and in RH" [W.-Y. Hsiang 3].

(10.55) [Barbosa, do Carmo].

Let Mm be compact and orientable, and

0: M' -- R"i+' an isometric immersion with constant mean curvature 0 0. Then 0 is V-stable if and only if c(M) is a Euclidean sphere.

157

ANOTHER REPORT ON HARMONIC MAPS

473

Consequently, the examples of (5.30) and (5.32) are not V-stable (m = 2); nor (for m > 3) are those of (10.54 (4)). (10.56) [Moore 1]. If Mm(c) is a compact real space form minimally immersed in S2m-' then the image is either totally geodesic or flat. (10.57) [Ejiri 1] has constructed isometric immersions of S3(4) in S' which are minimal, and totally real with respect to the invariant almost complex structure on S6. [Mashimo] has produced other isometric minimal immersions of S3(-,/[3k(2k+2)])

in S2" fork>3. (10.58) A submanifold M of S"-' is minimal if and only if the cone over M with vertex 0 is a minimal variety of l1".

(10.59) [Simons].

If 0: (Mm,g) --+ S" is a minimal immersion, then

V-index (o) 3 n - m,

V-nullity (0) ,>(m + 1) (n - m),

with equality only when (Mm, g) is a great m-sphere of S". Gauss maps

Various constructions analogous to that of (2.35) play a significant role.

(10.60) [Ishihara 4]. Let 0: (M, g) - S" be an isometric immersion of a surface. Define its Gauss lift q : M -+ Q(S") = SO (n + 1)/SO (n - 2) x SO (2) by ¢(x) = (gi(x), dq5(7 M)). Then 0 has parallel mean curvature if and only if is harmonic. (10.61) Let 0: (Mg) -+ R' be an isometric immersion, and ST 1(M) -+ M its normal unit sphere bundle, with standard bundle metric. The spherical Gauss map vo: ST'(M) -+ S"-' assigns to each vector in ST1(M) its translate to the origin in R".

Then [Rigoli 1], [Jensen, Rigoli 2]: v. is harmonic if and only if 0 has parallel mean curvature and conformal second fundamental form; that is, if we define f'c-W(O2cb_'T*(N)) for VE'(T'(M)) by

Q°(X, Y) = ,

then we require (fiPW> _ )2 for some smooth function .l: M-+ R.

The same assertion is valid [Jensen, Rigoli 1] for the spherical Gauss map vo: ST'(M) -+ ST(S") of an isometric immersion 0: (M,g) -+ S". And there is a version for general isometric immersions 0: (M, g) ---+ (N, h). (10.62) Let (M, g) be a Riemannian surface and 0: (M, g) -+ 1k" an immersionthough not necessarily an isometric immersion. [Jensen, Rigoli] have extended the results of (2.35) to this situation as follows. Consider ¢ as a map from (M, g) to C", using the inclusion li" c C", and define the complex Gauss map y': (M, g) - P"_1(E) by y, (x) = [00], a complex line in C". Then (1) y' factors through the quadric Qn_2 c Pn_1(C) if and only if 0 is conformal, (2) yc is harmonic if and only if 0 has parallel mean curvature, (3) yc is antiholomorphic if and only if 0 is harmonic.

158

474

J. EELLS AND L. LEMAIRE

(10.63)

Let (Mg) be

a Kahler manifold with dime M = m, and let

0: (M, g) -+ R" c C" be an isometric immersion with complex Gauss map yo': (Mg) -- Gm(C") c S(AmC"). Then (1) 0 is harmonic (equivalently, (1,1)-geodesic by (4.46)) if and only if yo is antiholomorphic ; (2) 0 is (2, 0)-geodesic if and only if y' is holomorphic [Rigoli, Tribuzy].

(10.64) [Rigoli 2] (following [Bryant 3] in dimension 2).

space with quadratic form

Let R"n" be the Lorentz

n+,

xv + Y, x;,

<x, x>

J-1

and `P+ _ {x: <x, x> = 0, x° > 0} the associated light cone. Define Gk(R1" n+1) the Grassmannian of k-spaces in W n+l on which < ,) has signature s. Then Gk(RI,n+l) is an open orbit in Gk(P"+2) with invariant pseudo-Riemannian metric [J. Wolf]. For any function p2: S" -+ R(> 0), let h,, = p2h, where h is the Euclidean metric of S". Then the embedding (S", h,,) -+ 2+ given by y --+ #(y) (1, y) is an isometry. Now let 0: (Mm, g) . (S", h) be an isometric immersion, and compose it with the embedding of S" in 2+. We define the conformal Gauss map 1,.: M -+ G,°,_m(l8' "+') by

F ,(x) = (d0(x) T(M))1, an (n - m)-space in l8' "+' with signature 0. Then Idrol2 =

2 I

>(1,0)+(2-m)(o,) A,, = f(0) Construct the functional

z(0) m

g.

rM

f, I dr,51m dx

W(0) = m

(jdFjm_2 dF¢) = 0. Thus on isometric immersions 0. The extremals are solutions of div for m = 2, 0: M -+ S" is an extremal of W if and only if its conformal Gauss map is harmonic. These extremals are called Willmore surfaces. For m > 3, IF, is harmonic if and only if 0 is pseudo-umbilical ( = 0) and has parallel mean curvature.

11. Non-compact manifolds

Throughout this Section we shall suppose that (M, g) is complete and noncompact. When m = 2, completeness is often not a necessary hypothesis on domains,

because any Riemannian surface (Mg) is conformally equivalent to a complete one-and the energy functional is a conformal invariant. Harmonic functions in 2P(M,R)

We consider now the existence of harmonic functions 0: (M, g) -. R in various classes. Most of these results can be cast in the more general context of subharmonic functions. See [Karp 1, 2, 3], [Li, Schoen]. (11.1) [Yau 3].

For I < p < oo any harmonic function in .P(M, l) is constant.

That was proved by [Andreotti, Vesentini] when p = 2, using Stampacchia's inequality.

159

ANOTHER REPORT ON HARMONIC MAPS

(11.2)

475

[Karp 1] has sharpened (11.1) in several ways, including the following.

If 0: (M, g) -- R is non-constant and harmonic, then for p > I and any nondecreasing function F(r) satisfying f ' dr/rF(r) = oo, 11M sup r-co

2

r F(r)

JcIPdx = oo

J D(x,,r)

and lim inf2 r-

0

r

1cIPdx = co J D(x,,r)

for all xo E M. (11.3) When 0 < p < 1, the situation is more complicated. There are examples of manifolds (M, g) supporting non-constant harmonic L'-functions (Chung), and LPfunctions for certain 0 < p < 1 (Sullivan). [Li, Schoen] give sharp curvature conditions to ensure that every harmonic function in 2P(M, R) is constant (0 < p < 1); see also [Li].

(11.4)

If Ricci(M,") > 0, then every harmonic function in .P(M, III) is 0

(0

(a) M is simply connected and Riem(M,') < 0; or (b) there are constants a < 0 and c > 0 such that - oo < a < Ricci(M, a) and c < Vol D(x, 1) for all x e M. Then any harmonic function in 21(M, EJ) is constant (0 < p is due to [Greene, Wu].

1). The case p >, 1 in (a)

Bounded harmonic functions (11.6) Natural extensions of Liouville's theorem should imply that any bounded harmonic function on (M,g) is constant; we examine supplementary hypotheses to ensure that.

(11.7) [Yau 2]. constant. (11.8)

If Ricci(M,9'

0, then any bounded harmonic function on (M, g) is

More generally [Donnelly]: if Ricci(M,9) > 0 outside a compact set, then the

vector space of bounded harmonic functions is finite dimensional. [Li, Tam] have produced manifolds with the same curvature restriction on which that dimension is greater than 1. (11.9) As an application of his Harnack principle, [Moser] has shown that if the Riemannian structure g on P' satisfies the uniformity condition

cYin

gy(Y,Y)
for each ye P"` and Ye Ty R' (c, Ce P(> 0)), then any bounded harmonic function 0: (Pm,g) -> P is constant.

160

476

J. EELLS AND L. LEMAIRE

On the other hand, simply connected manifolds with - oo < A < B<0 support non-constant bounded harmonic functions [Anderson], [Sullivan], [Anderson, Schoen]. Compactify M by adjoining the (m- 1)-sphere at infinity M(oo) (that is, the space of asymptotic classes of geodesic rays on (M,g)). Then M(00) has a C-manifold structure (a2 = B/A). There is a unique solution to the Dirichlet problem for harmonic functions on (Mg) with given continuous boundary values on M(oo). (11.10)

[Anderson] has applied this to show that there are proper harmonic maps from (M,g) onto the unit disc in R' inducing a homeomorphism of M(co) onto S'"-'. [Anderson]'s proof uses Perron's method, resting on the construction of suitable

barrier functions (that is, subharmonic functions f: (M, g) -> l(,< 0) for each xoeM(oo), with 0). [Sullivan]'s proof involves probabilistic potential theory. [Anderson, Schoen) have exhibited a natural homeomorphism of M(oo) onto the Martin boundary of (M, g). [Kasue] (and the references therein) give a general condition on the sectional curvature of a manifold, to ensure the existence of bounded harmonic functions. [Toledo]. Let (M, g) be a manifold having an isometric discrete principal action with compact quotient. Then its space of bounded harmonic functions is either one or infinite dimensional. Finite energy

(11.11) A characterisation of manifolds admitting non-constant harmonic functions of finite energy has been obtained by [Sario, Schiffer, Glasner]. If such a 0 exists on (M, g), then there is also a non-constant bounded harmonic function with finite energy. (11.12) If either Ricci"-9' >, 0 [Greene, Wu 1] or Vol (Mg) < oo [Schoen, Yau 1], then any harmonic function 0: (M, g) -+ f18 with E(q) < oo is constant. There are manifolds supporting non-constant bounded harmonic functions, but for

which every non-constant harmonic function has infinite energy [Sario, Schiffer, Glasner]. Holomorphic functions (11.13) Suppose now that (M, g) is a non-compact Kahler manifold. Then the real and imaginary parts of any holomorphic function 0: (M, g) -+ C are harmonic.

(11.14) [Yau 3].

Any holomorphic function in Y"(M, C) is constant (0 < p < co). Note that p ranges over all (0, oo), in contrast to the situation described above for harmonic functions. Also, (11.2) applies to this situation for all p > 0 [Karp 1]. And, from (11.7), if 0, any bounded holomorphic function is constant. (11.15)

There are non-compact Kahler manifolds on which every holomorphic

function is constant [Greene, Wu 2]. On the other hand, every Stein manifold is Kahlerian, and has enough holomorphic functions to separate points. (11.16) [Grauert]. Every Stein manifold supports a strictly plurisubharmonic function (4.32). [Greene, Wu 2]: a Kahler manifold is Stein if it supports a smooth strictly convex exhaustion function.

161

477

ANOTHER REPORT ON HARMONIC MAPS

Harmonic diffeomorphisms (11.17) If q5: 92.. 182 is an injective harmonic map, then it is affine. By way of contrast, O(z) = ez is holomorphic and surjective onto R2\0, with J,, > 0. In fact, let 0: R2 - R2 be harmonic and have only branch points. Then (i) if 0 omits one point, then ¢ factors through exp; (ii) if 0 omits more than one point, then it is constant [Sealey 1].

(11.18) There is no harmonic diffeomorphism of {z e C : zj < 1 } to C. The annulus {z E C : I < Izl < b < oo} cannot be mapped by a

harmonic

diffeomorphism onto a domain in C containing the complement of a disc (Bers). (11.19)

J. C. Wood has observed that the map 0: l3

183 defined by

g5(x, y, z) = (x3 - 3xz 2 + yz, y - 3xz, z)

is a harmonic homeomorphism of R3 to R3 whose Jacobian 3x2 vanishes on the yz-plane. Thus 0 is not a harmonic diffeomorphism. In another direction, let U c R3 be a domain and f: U -* R a harmonic function with 0 = Vf: U -). R' injective. Then J, A 0 on U [Lewy 2]. Existence theorems

We first discuss certain extensions of the existence theorems of Section 3 to noncompact domains. (11.20) [Burstall 1]. Let (N, h) be a compact manifold with contractible universal cover and assume that it has no minimising tangent map (3.26) of 98` for 3 < l < m, where m = dim M. Then any map 0,: (M, g) -+ (N, h) of f rite energy is homotopic to an energy minimising harmonic map.

If m = 2, it is sufficient to suppose that N is compact and ir2(N) = 0. Ditto if

m=3andn=2, as in (3.35). 0, the hypothesis of (11.20) is satisfied. That case is due to (11.21) If 0, [Schoen, Yau 1]. Moreover, if Vol (M, g) < co and (N, h) is compact with Riem(' then

if 0o, g1: (M, g) - (N, h) are homotopic harmonic maps of finite energy, then they are smoothly homotopic through harmonic maps of finite energy; (ii) if Riem(^'-h' < 0 and ¢: (Mg) (N, h) is a harmonic map offinite energy, then it is the only such map in its homotopy class, unless c(M) is contained in a geodesic of (N, h). (I)

We also have : (iii) if Vol (M, g) < oc and N is simply connected with Riem(N.h' '< 0, then any

harmonic map 0: (M, g) -+ (N, h) of finite energy is constant.

0 and if (11.22) APPLICATION. [Schoen, Yau 4] also show that if 0: M -+ N is a map for which there are compact sets K c M, L c N and a E Hk(M) such that 0 I,,,\K is null homotopic and ¢,,(a) 0 0 in Hk(N, N\L), then 0 is homotopic

162

478

J. EELLS AND L. LEMAIRE

to a map of finite energy. They make several applications to compact group actions; for example,

with M, N and 0 as above, if Vol (M, g) < co and Riern", " < 0, then dimIso(M,g) <, (m-k)(m-k+ 1)/2; (a)

(b)

any finite smooth group action on a homotopy torus is equivariantly homotopic

to a linear action on the standard torus. (11.23) The existence results for maps of surfaces can also be extended to the case where (N, h) is complete and non-compact, provided some conditions are imposed : (11.24) Uniformity condition. There are two positive constants c and C such that any point of N is in the domain of a coordinate chart 0: V -> Rn whose image is the unit disc, and cI dO(y)1'12a^ '< g&(Y, y) '< CI d9(y) y12

for any ye V and Ye 7,(N). (11.25) Growth condition. There is a function y : R(,> 0) - R(>, 0) with lim,_a y(r) = oc such that for some point yo E N the geodesic disc D(y, y(dist (yo, y))) is contractible for every y e N. (This is an improvement of condition (5.2) of [Report].)

(11.26) The uniformity condition was introduced by Morrey; in case dimM = 2 it guarantees that a minimising L; map is smooth, as in (3.19). The growth condition prevents the maps of a minimising sequence moving away

to infinity. This could happen because there are non-simply connected convex supporting

complete manifolds. By (3.31), the elements of their fundamental groups are not represented by closed geodesics. For instance, the quotient of the Poincare disc by a suitable group of translations is convex-supporting. (11.27) [Burstall 1] (improving on [Lemaire 6]). Let (M,g) be a surface and (N, h) a manifold satisfying the uniformity and growth conditions. Given any map V : (M, g) -+ (N, h) of finite energy, there is a harmonic energy minimising map 0: (M, g) -- (N, h) of finite energy such that on each compact set, they induce the same conjugacy class of homomorphisms on fundamental groups. If n2(N) = 0, then 0 is homotopic to v. The proof uses a minimising sequence and Morrey's regularity theorem.

Maximum principles (See also (12.10)-(12.13))

A real valued bounded function on (M,g) may not have a local maximum. The following result is a substitute, due to [Omori], with refinements added by [Yau 1]. (11.28) Let - oo < a ,< Ricci(M,) and f: M- - R a junction with sup f < co. Then there is a sequence (xk) c M such that

(i) limf(xk) = supf (ii) lim ldj(xk)I = 0

(iii) lim -Aj(xk) 6 0.

163

ANOTHER REPORT ON HARMONIC MAPS

479

(11.29) That theorem has been applied to various situations ; in particular, to the Schwarz lemma for holomorphic maps [Report (9.14)] ; to harmonic maps of bounded dilatation [Report (5.10)] ; and to minimal immersions inside a nondegenerate cone of Euclidean space [Omori]. An extension of the latter by [Baikoussis, Koufogiorgos] asserts that if (M, g) satisfies - oo < a < Ricci(', 91, then there is no harmonic map 0: M -> 1l' with idol >, c > 0 and q'(M) inside a nondegenerate cone of R'.

(11.30) [Karp 3].

Let (M,g) satisfy 1im sup 11og Vol D(xo, r) < oo, ;:2

and let f: M -+ TI be a function bounded from above. Then inf - Of < 0. (11.31) [Tolksdorf 1]. Let Mo be a domain in M and o: Mo -D(yo, r) c (N, h) a non-constant continuous map, where r < 1 -,1 B. If 0 is weakly harmonic in Y '(M,, N), then dist (O(x), y() < sup dist (c(y), yo) yEaM,

for all x e Mo.

Liouville theorems for maps

By a Liouville theorem for maps, we mean here a statement to the effect that a harmonic map with suitably small image is constant. In fact, the measure of smallness is often one of the conditions appearing in the regularity theory of Section 3. (11.32) If (M,g) has Ricci(M'111>,0 and (N,h) is simply connected with Riem(" ") < 0, then any harmonic map 0: (M, g) --> (N, h) with relatively compact image is constant [Cheng 2].

(11.33) [Karp 1] has replaced the boundedness hypothesis by a condition on the rate of dispersion of 0, as follows. Let F: T(>, 0) OI(>00) be as in (11.2), and let 0: (M, g) (N, h) be harmonic, 0. If 0 is not constant, then for all where N is simply connected and p > 1 and for any xo e M we have

lim sup r-.oc

distp(q(xo), O(x)) dx = oo ;

1

rs F(r)

JD(xo, r)

and

lim inf 2 r-.m

r

dist9(c(xo), ¢(x)) dx = oo. J D(x,,r)

(11.34) In another direction, [Karp 2] has shown that if - oo < a < Ricci"', 9), 0 and m < n, then (M, g) admits no bounded (N, h) is simply connected with Riem("' minimal immersion in (N, h).

The curvature condition on the domain cannot be deleted, for [Jones] has constructed a bounded holomorphic curve in Ca.

164

480

J. EELLS AND L. LEMAIRE

Ricci(M.F' and Riem"' < B < co for some B > 0. If (11.35) Let - oo < a < (M, g) --> (N, h) has image in a geodesically small disc D(y, r) (3.14), then [Xin 6] uses (11.28) and the Hessian comparison theorem [Greene, Wu 3] to prove

sup Ir(¢)I tan s/Br >, 2 ,/B infe(r).

(11.36)

(11.37) APPLICATION. Let ¢i: (M,g) -> l be a proper isometric immersion with parallel mean curvature, and y.: M G.(Rn) its Gauss map. Take B = 1 if n = m + 1 and B = 2 otherwise. If yo(M) c D(y, r) with r < 2n.\/ B, then 0 is harmonic. Indeed, by (2.35), y, is harmonic, so (11.35) implies that inf e(y,) = 0. But dy" = fl (o), so that Schwarz's inequality gives Ir(O)I2 < fl(0)I2, so inf lr(q)I2 = 0. But e(y¢) = 2If(cb)I2. Ir((/J)12 is a constant for immersions of parallel mean curvature, whence r(q5) = 0. If Ricci(M,°' > 0, then y. is constant by (11.38) below. Thus 0 maps M into an m-dimensional subspace of l ".

(11.38) [Choi 1], [Q.-H. Yu]. Let - oo < a ,
small disc D(y, r), then e(q) is bounded by a constant depending on a, B and r. If 0, then 0 is constant. (11.39) [Hildebrandt, Jost, Widman]. Take (R', g) as in (11.9) and any (N, h) with Riem", " < B < oo. If 0: (11", g) -- (N, h) is a harmonic map with q(M) contained in a geodesically small disc, then 0 is constant. [Kendall 3] used stochastic methods to prove that if (M, g) is any manifold on which every bounded harmonic function is constant, and 0: (M, g) -> (N, h) a harmonic map with image in a geodesically small disc, then 0 is constant.

That is a common generalisation of (11.38) using (11.7) and (11.39) using (11.9).

(11.40)

In contrast, [Aviles, Choi, Micallef] have established the following

existence theorem: Let (M,g) be simply connected and satisfy A < Riem"M'g' < B < 0. If M(oo) is the sphere at infinity of M and yr: M(ao) --+ (N, h) a continuous map with image in a geodesically small disc D(y, r), then there exists a unique harmonic map 0: (M, g) - (N, h) such that 0 I w and qS(M) c D(y, r). (11.41)

Constancy of harmonic maps can also be ensured by bounds on the

energy, as follows. Let Ricci'M'D' >, 0,

0 and let 0: (M, g) -+ (N, h) be a harmonic map with

E(c) < co. Then 0 is constant [Schoen, Yau 1] and [Hildebrandt 1]. (11.42) Give Rm a complete metric of the form g = f 2 go, where go is Euclidean. Suppose that m >, 3 and 8/8r(r J(x)) > O for all x, where r = Ixl. If 0: (Rm, g) -a (N, h)

is a harmonic map of finite energy, then ¢ is constant [Sealey 3], following [Garber, Ruijsenaars, Seiler, Burns] for the case f = 1.

165

ANOTHER REPORT ON HARMONIC MAPS

481

The condition on f means that the mean curvature vector of each geodesic sphere in (R', g) centred at 0 points to the interior. For example, we could take for (R', 9) a real or complex hyperbolic space form. However, (11.42) is false for m = 2: there are many non-constant conformal maps 0: R2 --+ S2 with E(0) < oo. See [Xin 5] for a restriction on the curvature of (M, g) producing a conclusion like that of (11.42). (11.43)

The finiteness of the energy in (11.42) can be replaced by various

growth conditions. For instance [Hu], let F: R(,> 0) -+ R(>, 0) be as in (11.2) and let 0: (R', go) -* (N, h) be a harmonic map with f Bm e(O)/F(r) dx < oo. If m > 3, then 0 is constant. (11.44) The following Liouville theorems correspond to the regularity results of (12.15) and (12.19) below.

(11.45) [Giaquinta, Soucek], [Schoen, Uhlenbeck 3]. Let S+ denote the closed unit hemisphere in Rn+a Every harmonic map 0: Rm -+ S+ is constant for m < 6. (11.46) [Schoen, Uhlenbeck 3].

Define d(n) by

d(3) = 3 and d(n) = min ((n + 2)/2,6) if n > 4. Then for m <, d(n), any E-minimising harmonic map 0: Rm -+ S" is constant. Define d(3) = 2 and d(n) = min ([n/21, 4) if n >, 4. Then for 2 5 m < d(n), every E-

stable harmonic map 0: R' - Sn is constant. (11.47) [Solomon]. Let Si-2 be a great sphere of codimension 2 in Sn (n ? 2) and V a neighbourhood of Sn-2. Any energy minimising map 0: I18m --+Sn\V is constant.

(11.48) [Xin 7]. If (M,g) is complete with moderate volume growth and G/H a compact irreducible Riemannian homogeneous space whose first eigenvalue A < (2/n) Scalo"H, then any stable harmonic map 0: (M, g) -- G/H is constant. Analogous results under different hypotheses can be found in [Wei 1, 2] and [Xin 7]. See [Meier 5] for asymptotic limit theorems for harmonic maps with geodesically small range. (11.49) A map 0: (M, g) --> (N, h) has bounded dilatation if there exists a number ...j /A2 < K2, where K such that for each x e M, either dq (x) = 0 77

are the positive eigenvalues of O*h at x ([Report (5.9)]). The following results have been derived using stochastic methods. (11.50) [Kendall 2]. Suppose (M, g) supports no non-constant bounded harmonic function. Let (N, h) be simply connected and such that

-oo
(11.51) [Elworthy, Kendall). Let n: (P, k) --. (M, g) be a Riemannian submersion with totally geodesic fibres. Suppose that the fibres are compact or have Ricci > 0. If (N, h) is simply connected and B < 0, and ty : (P, k) -+ (N, h) is a harmonic 16

ELM 20

166

482

J. EELLS AND L. LEMAIRE

map with bounded dilatation, then V is constant on the fibres-and thereby factors through 7t to give a harmonic map 0: (M, g) -+ (N, h). Harmonic morphisms (11.52)

Let U c 68m be a domain, (N, h) an oriented surface and 0: U -> N a

submersion. Let Gm-2(Fm) be the Grassmannian of oriented (m - 2)-planes in Rm. To we associate a Gauss map 70: U _ Gm-2(R7) by setting yO(x) = Tz(U). Using the identification of Go .-,(R'4) with the complex quadric Qm_2, [Baird, Wood] obtain these results. Let 0: U -* N be a submersion with connected totally geodesic fibres. (a) If 0 is horizontally conformal, then y4 is horizontally holomorphic in the sense that dyO o J" = Jo dyO (where J" is the operator induced by the complex structure of N on the horizontal distribution); in particular, yO is harmonic. (b) yO factors through 0 to give a ±holomorphic map from N to Qm-2. (11.53) The following completely explicit descriptions were obtained by [Baird, Wood] : Any harmonic morphism from 4183 to a manifold (N, h) is either (a) constant; (b) a harmonic map to R or S1; (c) the composition of an orthogonal projection 683 -+ 682 followed by a conformal

map; (d) a homothetic map R3 --+ R3 or If83 -+ R3/F, where F is a discrete isometry group. Any harmonic morphism from S3 to a manifold (N, h) is either (a) constant; (b) the Hopffibration S3 -* S2 (see [Report (8.4)]) followed by a conformal map,

up to an orthogonal transformation of R; (c) a homothetic map S'-S' or S3 --+ S3/I'. Holomorphic maps

The following is a complex analogue of (11.41).

(11.54) [Sealey 3], [Schoen, Yau 1]. Let (M, g) and (N, h) be complete and 0 and HBRiemc".n) <, 0. Then for any holomorphic map Kahler, with 0: (M, g) (N, h), e'(0) is subharmonic. If E'(q5) < oo, then e'(0) is constant. If 0, then 0 is constant. (11.55) Let (N, h) be a Riemann surface with positive curvature. If 0: C -> (N, h) is harmonic and a local E-minimum with Jacobian JJ > 0, then 0 is holomorphic [de Bartolomeis].

(11.56)

Let 0: (M, g) --+ (N, h) be a holomorphic map between Kahler manifolds,

and f: (N, h) -+ 11 a plurisubharmonic function (4.32). Then it is elementary that f o 0: (M, g) -+ R is subharmonic. (11.57) [Takegoshi]. Let (M,g) be a Kahler manifold with dime M >, 2. Suppose there is a function f: (M, g) -+ 68 such that IVf I < I on M and f is (m -1)-

167

483

ANOTHER REPORT ON HARMONIC MAPS

plurisubharmonic, strictlyso somewhere. Ifa (1, 1 )-geodesic map (4.43) c : (M, g) --+ (N, h)

into a Kahler manifold has E(0) < cc, then 0 is constant. Any harmonic function with finite energy on (M, g) is constant. (11.58)

Let (M, g) be a non-compact Kahler manifold with moderate volume growth

(that is, there exists xo E M and a function F as in (11.2) such that lim sup r-. c,

2

1

r F(r)

Vol D(xo, r) < oo ).

Suppose that (N, h) supports a strictly plurisubharmonic function. If 0: (M, g) -+ (N, h) is holomorphic with relatively compact image, then 0 is constant. [Karp 1]. For instance, we could take for (N, h) a non-compact Kahler manifold (1) (2)

which is simply connected and Riem"''I < 0; or HBRiem", 'I >, 0, and >0 outside a compact set.

Minimal surfaces in R3

The standard reference for this extremely rich topic is [Nitsche]; also [Meeks 2, 3].

(11.59) We follow tradition in calling a harmonic isometric immersion 0: (M2,g) -+ (183 a minimal surface, where we emphasise that (M2,g) must be

complete. Such surfaces are characterised as the extremals of the area (= volume) functional (2.21).

(11.60) Classically, only three such embedded surfaces* were known: the plane, the catenoid, and the helicoid. Recently, a new one was found, a minimal surface of

genus one with three ends. It was constructed by [Costa] by introducing the Weierstrass p-function in the Weierstrass representation formula for surfaces; by studying its symmetries [Hoffman, Meeks 1, 2] proved that it is embedded. Subsequently they obtained other surfaces of higher genera and with various end structures. (11.61) Concerning minimal immersions, we have some indications on their possible behaviour. If 0: (M,g) -> R3 is an isometric immersion then its Gauss map 0: (M,g) S2 is antiholomorphic if and only if 0 is harmonic (2.35); is holomorphic if and only if O(M) lies in a sphere or plane [Hoffman, Osserman] ; and is harmonic if and only if 0 has constant mean curvature (2.35).

(11.62) If M is a non-planar immersed minimal surface in 183, then its Gauss map cannot omit 5 points of S2. That was first proved by [Xavier] for 7 points, then improved to 5 by [Fujimoto].

It extends a theorem of [Osserman 1] that the Gauss map cannot omit a set of positive logarithmic capacity. There are minimal surfaces whose Gauss maps omit 4 points. (11.63) [Jorge, Xavier 1] used the Weierstrass representation formula to produce a rather surprising non-planar immersed minimal surface between two parallel planes

in 183. In particular, one of its components is a non-constant bounded harmonic function on the surface. *of finite topological type.

16-2

168

484

J. EELLS AND L. LEMAIRE

(11.64) A minimal surface in R3 has zero V-index with respect to compactly supported variations (2.23) if and only if it is a plane in 683 [do Carmo, Peng], [FischerColbrie, Schoen].

That assertion is false for minimal immersions ¢: M -r 68", as can be seen by examining holomorphic curves in C2. (11.65) Replacing P3 by a manifold, [Fischer-Colbrie, Schoen] extend (6.26) to the non-compact case as follows : Let (N, h) be an oriented 3-manifold with Sca1M 'n' >, 0, and M an oriented noncompact V-stable minimal surface in (N, h). Then (M, g) is conformally equivalent to C or to C\{0}. In the latter case, M is totally geodesic and Scab' '°) = 0 along 0. If, moreover, Ricci'" ) > 0, then M is totally geodesic and the normal Ricci curvature is zero along 0. (11.66) In the spirit of (6.23), [Schoen, Yau 7] use the above results to prove that any complete non-compact 3-manifold with positive Ricci curvature is diffeomorphic to P. They also show that if N is a 3-manifold such that ttl(N) contains the fundamental

group of a compact Riemann surface of positive genus, then N does not carry any complete metric of positive scalar curvature.

[Meeks, Simon, Yau] use (11.65) to show that if M is a compact minimally embedded surface in an orientable 3-manifold (N, h) with Ricci('-') >, 0, then (N, h) is isometric to M x R. (11.67) [Pitts, Rubinstein 2].

Every non-compact 3-manifold with constant negative

curvature and finite volume contains an embedded minimal surface of finite area. This is an application of the minimax procedure mentioned in (5.22). See also [Uhlenbeck 3]. (11.68) The total absolute curvature of an isometric immersion 0: (M, g) -> P3 is the left hand member of the identity

f.M

IKMI dx = V(yo),

where KM is the Gaussian curvature and V(y..) the volume functional applied to the Gauss map. (11.69) Let 0: (M, g) - R3 be a minimal immersion. The following are equivalent: (a) 0 has finite total absolute curvature; (b) c(M) is conformally equivalent to a closed surface Mo with a finite number of

points removed; (c) its Gauss map extends to a meromorphic function on M0; (d) 0 has finite V -index. The equivalence of (a) and (b) is due to [Chern, Osserman]; that of (a) and (d) to [Gulliver 3] and [Fischer-Colbrie]. Furthermore [Gulliver, Lawson], in that case M has finite topological type and quadratic area growth. If for some compact subset F, M\F is minimal and V-stable, then M has finite topological type, quadratic area growth, and finite total absolute curvature.

169

ANOTHER REPORT ON HARMONIC MAPS

485

Surfaces of constant mean curvature in 3-manifolds (11.70) Applying the method of (5.32), [Kapouleas] has shown that if p >, 0, m > 3, or p >, 2, m = 2, there are infinitely many properly immersed surfaces of

constant mean curvature in UF3 of genus p with m ends. (11.71) Moreover, [Korevaar, Kusner, Solomon] have obtained the following restrictions on properly embedded surfaces M of constant mean curvature in l . (a) If M is contained in a cylinder, then M is a Delaunay surface (5.32). (b) If M has finite topological type, then each end exponentially approaches a Delaunay surface. (c) These Delaunay surfaces satisfy a balancing condition as in (5.32). [Hoffman, Osserman, Schoen] use a result of [Fischer-Colbrie, Schoen] to prove the following: (11.72) Let 0: (M, g) -+ l3 be an oriented surface isometrically immersed with constant mean curvature. If the image y,(M) c S2 of its Gauss map lies in an open hemisphere, then c(M) is a plane. If y,,(M) lies in a closed hemisphere, then O(M) is either a plane or a right circular cylinder.

(11.73) Say that an immersion 0: (M, g) --* 183 of constant mean curvature H (= Iz(&)I/2) is stable if

f IVfl2-(4H2-2KM)f dx

0

for each compactly supported variation f: M -+ R such that f M fdx = 0. (11.74)

[Palmer] and [da Silveira] have shown that a V-stable immersion of

constant mean curvature in 183 is a Euclidean sphere or a plane. There are analogous statements for immersions into I H3. [Meeks 4] has used (11.74) to prove that a foliation of R3 by surfaces of constant mean curvatures is by parallel planes. Minimal graphs (11.75) For any map 0: M -> N its graph map D : M --> M x N is defined by c(x) = (x, O(x)). With the induced metric cp*(gxh) =g+cb*h, -0: (M,g+c*h),(MxN,gxh)

is an isometric embedding. Let 02 = 7r2 0 D, where rr2: (M x N, g x h) --> (N, h) is the projection on the second factor. Then: (11.76) 1): (M, g+ ¢ *h) -+ (M x N, g x h) is harmonic (= minimal) if and only if (D2: (M, g+ cb*h) --> (N, h) is harmonic.

In particular, that is not equivalent to 0: (M, g) --> (N, h) harmonic. (11.77) [Morrey 3] has shown that a C'-solution of r(O) = 0 is smooth ; however, [Lawson, Osserman] have exhibited Lipschitz solutions which are not C' (in that case,

g+¢*h is not continuous; compare with (3.9)).

170

486

J. EELLS AND L. LEMAIRE

(11.78) If 0: (M, g) -- (N, h) is an isometric immersion, then g+0*h = 2g, so that (D2: (M, g+ ¢*h) -+ (N, h) is harmonic if and only if 0: (M, g) --> (N, h) is. (11.79)

The following isoperimetric inequality is due to [Heinz] for maps of R2

to R, to [Chern 1] and [Flanders] for maps of l' to R ; a simple geometric proof (involving foliations) is given in [Barbosa, Kenmotsu, Oshikiri]. The general case has been established by [Salavessa]. For any map 0: (M, g) - (N, h) with V'-r((D) = 0 (and hence fir(O)I = constant), we have the isoperimetric inequality Ir((D)l < Vol (OD)/Vol (D)

for each smooth compact domain D c M. A key step in the proof is to show that f r(b)I2 = diva Z on M, for a suitable vector field Z. (11.80)

Define Cheeger's constant b(Mg)of (M, g) by

E)(M,g) = inf{VOl(8D)/Vo1(D): D compact smooth domain of M}. For instance, El(M, g) = 0 if (M, g) is compact ; or if (M, g) = R' with a metric such that Ago < g < Bgo, where go is the Euclidean metric. (11.81) Application of (11.79) yields: Let (M,g) be a m a n i f o l d with 1)(M, g) = 0. Then t :

( M N,gx h)

has constant mean curvature if and only if it is harmonic. That equivalence may not be valid if 4(M,g) 0 0. [Salavessa] has constructed an explicit function ¢: RH2 R on the real hyperbolic plane whose graph map 0 has 1.

(11.82) [Salavessa] has also shown that for any (M, g) and (N, h), if 0: (M, g) (N, h) is an isometric immersion, then D has constant mean curvature if and only if c is harmonic; or equivalently if 0 is harmonic. A similar result holds if dim M = 2 and 0 is weakly conformal. (11.83)

Consider a map 0: 18m -. R' and its graph map $: (R",go+O*ho) - (a8'" x R ,go x ho)-

For n = 1, any solution of r(d) = 0 is totally geodesic for m <, 7; but not necessarily so for m >, 8. Bernstein showed that any minimal graph on R2 is planar; Almgren* did similarly for m = 3, 4; Simons for m = 5, 6, 7. [Bombieri, De Giorgi, Giusti] proved (11.83) in general. Their solution is based on the study of minimal cones and singularities of minimal submanifolds.

See [do Carmo, Lawson] for such a Bernstein theorem in RH". In higher codimension n > 1, (11.83) has no analogue: indeed there are non-affine minimal graphs for m = n = 2 (Osserman). However, it has been shown by Yau that any minimal graph of 0: R2 R" is conformally equivalent to C.

-

*and De Giorgi.

171

ANOTHER REPORT ON HARMONIC MAPS

487

Maps of Lorentzian manifolds (11.84) Let (M,g',m) be a Lorentzian manifold, where gl'm = g has signature (1,m), and let (N,h) be a compact Riemannian manifold (h positive definite). The Cauchy problem is to find a harmonic map 0: (M, g) -. (N, h) with the initial data 0

and do along a space-like hypersurface of (M, g). The condition of harmonicity is now expressed by a semi-linear hyperbolic system. (11.85)

The following existence and non-existence results have been obtained.

(i) Local existence for M = R x M0, where the restrictions g, = g txMo are positive definite and uniformly equivalent to a fixed go [Choquet-Bruhat]. (ii) Global existence when (M,g) = lv"' is the Minkowski plane [Gu 1]; see also [Choquet-Bruhat]. Unique global solution to the initial-boundary problem for a strip in 18',' [Gu 2]. (iii) Global existence when (M,g) = R'," is the Minkowski space and the Cauchy data are sufficiently small [Choquet-Bruhat].

(iv) Existence of weak global solutions for (M, g) = 68'' m and (N, h) = S provided the energy of the Cauchy data is finite [Shatah].

(v) In contrast, Shatah gave an explicit example of nonexistence for a map 0: P1.3 - S3, in which 0 develops a singularity in finite time. (11.86) [Bunting] and [Mazur] have proven the uniqueness of the solution to certain black hole models by transforming their equation into that of a harmonic map

into a pseudo-Riemannian symmetric space, and applying a uniqueness theorem there.

12. Manifolds with boundary (12.1)

In this Section (Mg) denotes a compact manifold with smooth (not

necessarily connected) boundary aM. Unless otherwise specified, (N, h) will have no boundary. When it does, we shall say that ON is convex (resp. strictly convex) if its second fundamental form is semi-definite (resp. definite), pointing inwards. Such a convexity hypothesis is required for the existence of harmonic maps. Indeed, if aN is not convex, we can find two points near the boundary which cannot be joined by a geodesic. (12.2) Let yr : OM -* N be a map extendible to M and denote by ' (M, N) the space of extensions of yr to maps M - N. We are led to two versions of the Dirichlet

Problem :

Given yr, is there a harmonic map 0: (M, g) -- (N, h) such that 0 Ia,N = yr? Can we

prescribe the component of 0 in c 6,(M, N) (in other words, its relative homotopy class)?

Regularity and existence (12.3) The regularity theory of (3.21)-(3.27) has been extended to the present situation by [Schoen, Uhlenbeck 2]. Assume (N, h) is compact. Suppose that yr E " (8M, N) admits an L;-extension to M, and that 0 e Ii(M, N) is an energy minimising extension. Then 0 is 162+a in a

172

488

J. EELLS AND L. LEMAIRE

neighbourhood of 3M; its singular set is closed, disjoint from 8M, and has Hausdorff dimension restricted as in (3.21) and (3.27). (12.4)

A new ingredient in their proof arises from the following theorem of

[J. C. Wood 2] ([Lemaire 2] in case m = 2); here the absence of hemispheres plays the role of the absence of spheres in (3.27). Let Dm be a flat disc in 18m (m > 2) and 0: Dm -+ (N, h) a harmonic map which is constant on 8D'". Then 0 is constant on Dm. (12.5)

As in the case of no boundary, (12.3) together with the control of the

R, action yields various existence results. In particular, a new proof of [Hamilton 1]'s theorem : (12.6) If N is compact, Riem"" ') < 0 and yu e '(aM, N), then every component of ',(M, N) contains a harmonic representative, which is an E-minimum. Furthermore, that harmonic map is unique in its component. (12.7) If (N, h) is non-compact and ¢ le, = yr has a solution [Jost 5].

(12.8)

0, then the Dirichlet problem

If dimM = 2, if N satisfies the uniformity condition (11.24), and if

w e 1°(8M, N) admits an L;-extension w to M inducing the homomorphism yr* on the fundamental groups, then there exists an energy minimising harmonic map ci : M -* N, continuous on M and smooth in the interior, such that 0 I aM = yi Ian., and 0* = yr,k. If

furthermore rc2(N) = 0, then every component of ',,,(M, N) contains such a map [Lemaire 5]. Note in particular that the existence of an L'-extension implies the existence of a continuous extension. The same method shows that if N is compact or satisfies the uniformity condition

(11.24) and growth condition (11.25), then the Neumann problem r(q) = 0 on M, 0, 0 = 0 on OM has a solution for every conjugacy class of homomorphisms from Tr,(M)

to n,(N). (12.9) As in (3.35), if dim M = 3 and N is a surface not homeomorphic to S2 or P2(R), then any component of W,(M, N) contains a harmonic representative.

(12.10)

If yr : OM -+ (N, h) has its image in a geodesically small disc D (3.14), then

the Dirichlet problem has a solution [Hildebrandt, Kaul, Widman]. That solution is unique.

Indeed, [Jager, Kaul 1] have established the following maximum principle. If 02: (M, g) -> (N, h) are two harmonic maps with image in a geodesically small disc D(y0, r) (with r < 2n \/B) and qB(t) =

((1-cosy1Bt)/B ifB>0 It 2 /2 ifB=0,

then the function 0 = 8(0,, cb2) : (M, g) -. D defined by B(x) =

ge(dist (0,(x), c2(x))) dist (y0, p2(x)))

cos (s/B dist (y0, 0,(x))) cos

satisfies sup, 0 < sup,,,, 6. In particular, if 0,I a,,,, = 021,M, then 0, = 02.

173

489

ANOTHER REPORT ON HARMONIC MAPS

Originally these existence and unicity results were obtained under the assumption that D(yo, r) is disjoint from the cut-locus of each of its points; [Jost 2] improved them to the present form, using condition (3.14). (12.11) Moreover, if VI(8M) c N, c D where No has smooth convex boundary, then y/(M) c No [Jost 4]. (12.12)

A priori estimates for boundary regularity have been obtained by

[Giaquinta, Hildebrandt], [Hildebrandt, Jost, Widman], [Jost, Karcher [Jost, Meier], [Meier 2], [Schoen, Uhlenbeck 2], [Sperner].

1,

2],

(12.13) Another maximum principle is due to [Jost 5]. Let Bo C B, C (N,h) be closed sets and it: B, -> Bo a retraction such that d'(it(x), 7r(y)) < dN'(x, y) for all

x, y E B,\Ba. If 0: (M, 8M) --* (B Bo) is an E-minimising L'-map with fixed boundary data, then 0 carries M into Bo a.e. Here is an application, in conjunction with (12.3): Let N, be a compact submanifold of (N, h) with strictly convex boundary, such that

No supports a strictly convex function. Then given w : 8M -+ No, there is an Eminimising harmonic map 0eT,,(M, N) with O(M) c No. When M is a surface, (12.13) also yields a simple proof of (12.8). See also [Sealey 5] and [Nishikawa]. Maps to a closed hemisphere (12.14)

Let S+ denote the closed unit hemisphere in I"+1 Note that it is a closed

disc in S" of radius pr = it',/B, and therefore just outside the scope of the hypotheses of (12.10). Further analysis of the conditions in (3.27) shows [Schoen, Uhlenbeck 3], [Giaquinta, Soucek] : (12.15) Any energy minimising map 0: (M, g) - S+ is smooth if m < 6; has at most isolated singularities for m = 7, and has a closed singular set of Hausdorff

dimension < m - 7 for m > 8. This immediately implies that for any smooth map yr: 8M -> S+, the associated Dirichlet problem has a smooth solution if m < 6, which is an energy minimising map. (12.16)

That result is sharp. Indeed, consider the discontinuous map 0: Dm - Sm

obtained by projecting Dm\{0} radially onto its boundary Sm-', then embedding Sm-' as the boundary of Sm by the isometry yr; in coordinates Rm n Dmax-+

\Ixl'01ESm C R

'

For m > 3 the map 0 is Li and weakly harmonic [Hildebrandt, Kaul, Widman]; it is energy minimising if and only if m > 7; it is unstable for 3 <, m < 6 [Jager, Kaul 3]. (12.17) Furthermore, the isometric embedding yr: OD' - Sm does not extend as a harmonic map from Dm to Sm for m > 7 [Baldes 2]. We stress that this assertion excludes the existence of any harmonic extensionnot just one in a given homotopy class, or one which is energy minimising. This is in fact the first counterexample to the first part of (12.2).

174

490

J. EELLS AND L. LEMAIRE

The proof uses the particular nature of the boundary data, and the fact that for nearby data (inside the open hemisphere), the problem has a unique solution (12.10).

That is used to show that any harmonic extension must be equivariant, which is impossible as we shall see in (12.26) below. The special nature of these boundary data is illustrated by the following example of [Eells, Lemaire 5]. There is a harmonic polynomial map yr : OD' S' of polynomial degree 5 which is homotopic to the identity map, and which has a harmonic (12.18)

extension 0: D8 --+ Se.

Maps to spheres (12.19) Restrictions on the singular set of maps from manifolds to spheres have also been obtained as follows :

[Schoen, Uhlenbeck 3]. If 0: (M, g) _+ S" is an energy minimising map, then dim Y. < m - d(n) - 1, where d(n) is defined in (11.46). In particular, if m < d(n), then is smooth.

[Solomon]. If 0: (M, g) -+ S" is an energy minimising map which omits a neighbourhood of a great sphere Sn-2, then ¢ is smooth. We stress once again the correspondence between these regularity results and the Liouville theorems (11.46) and (11.47). (12.20) Analogous restrictions on the singular set have been obtained by [Xin 4] for maps into compact irreducible homogeneous spaces; and by [Wei 2] and [Xin 5] for other special ranges.

(12.21) A detailed study of the number and arrangement of the (discrete) singularities of E-minima in ',,(D3, S2) is under way : [Almgren, Lieb], [Brezis, Coron,

Lieb 1, 2], [Hardt, Lin]. In particular, for fixed yi, the number of points of discontinuity of E-minima is bounded by a constant times E(V) [Almgren, Lieb]. Maps of the disc (12.22)

Wood's result (12.4) was much extended by [Karcher, Wood], including

the following. Let (Dm,g) be the closed m-disc (m 3 2) with a metric of the form g = f g(,, where go is the Euclidean metric. Assume that al(r(r-f(x)) >, 0 for all x e Dm (r = Ixl). If 0: (Dm, g) -+ (N, h) is a harmonic map with I aDm constant, then is constant on Dm.

In particular, this applies if D' is a small enough geodesic disc. For m >, 3, the proof uses a generalisation of the basic formula of [J. C. Wood 2] : if o: (Dm, go) --+ (N, h)

is harmonic, then

E(¢IaDm) = (m-2)E(c)+2

aDm

JIa,c12dx.

(12.23) If nm(N) 0, (12.22) implies that for w constant we cannot find any harmonic map in the non-trivial components of W,,,,(Dm, N).

By way of contrast, we remark that for m > 3, we have no example of homotopy classes of maps between closed manifolds containing no harmonic map, although we have little doubt that many should exist. For m >, 3, [Ratto 4] has constructed special symmetric metrics g on Dm such that the Dirichlet problem for harmonic maps 0: (Dm, g) -> Sn with aDm constant has non-constant solutions.

175

ANOTHER REPORT ON HARMONIC MAPS

491

(12.24) A related result of [J. C. Wood 2] is that if Am(a,b) _ {xeIlBm: 0 < a < IxJ S b < oo} is an annulus with a metric satisfying the condition of (12.22), then any harmonic map O: Am(a, b) --+ (N, h) such that 0 ISm-i(b) is constant and a, 0 ISm- a) = 0 must be constant. (12.25) Some idea of the existence problem can be obtained by consideration of the following restricted class of maps. On (Dm, go) (resp. on (Sm, ho)) take the radial coordinates (r, O), r E [0, 1 ], O e (resp. (R, O), R e [0, 7t], O e S "'1). A map Sm-1

S' is equivariant if it is of the form 0(r, 0) = (R(r), 19(0)). Such a map 0 can be harmonic only if 0 is a harmonic k-homogeneous polynomial map from 0: Dm

Sm-1 to Sn-1. (12.26)

Following [Jager, Kaul 3], consider such a map 0: Dm -+ Sm, with

0(6) = 0. Then 0 is the restriction to Dm of an equivariant harmonic map u: Rm -+ Sm, given by yr(r, 0) = (R(r), 0).

For m = 2, R = 2 arc tg (c r), with c > 0. For m

3, limr-. R(r) = 7t/2; and

furthermore for 3

m < 6, the function R(r) oscillates around its limit, whereas for m >, 7 it is monotonically increasing to 7c/2. Therefore, for 3 < m < 6, there exist infinitely many equivariant solutions to the Dirichlet problem R(1) = 7t/2, whereas for m > 7 there are none.

The proof of [Jager, Kaul 3] is based on the stability theory of solutions of ordinary differential equations around their critical points. (12.27)

More generally, let yi: OD' -> Sn-' c Sn be a harmonic k-homogeneous

polynomial map to an equator of

Sn.

If m < c = 2+2(1 + \/2k), then w admits a

harmonic extension 0: Dm --+ Sn; if m > c, then w does not admit such an extension. This result of [Eells, Lemaire 5] is an extension of those of [Jager, Kaul 3] and [Baldes 2], and leads to example (12.18) above. (12.28)

The above results on maps from manifolds to hemispheres admit various

extensions to maps into ellipsoids and other rotationally symmetric manifolds : [Baldes 2], [Helein], [Karcher, Wood], [Tachikawa 1, 2]. (12.29) Let 2 < n 5 m-1 and define 0o e 1i(Dm, S") by ¢o(x', x") = x'/Ix'I for Ohm-n-1 Then 0a is 0-homogeneous, and minimises E amongst all (x', x'e 118n+1 x

&y2 (Dm, Sn) with 0 _ ¢o on DD' [Coron, Gulliver]. The case n = m -1 is due to Lin, and that of m = 3, n = 2 to [Brezis, Coron, Lieb 1, 2]. Furthermore [Coron, Gulliver], for any odd map yr: S2 -+ S2 (that is, yr(-x) _ - v/(x) for x e S2), 0o minimises E amongst all o E 2 (D3, S2) with = w on OD3. A key idea in the proof of (12.29) involves averaging E over the Grassmannian G3(Rn+1) with respect to its invariant volume dP. There is a constant c(n) such that for any 0 e .Pi(Dm, S') we have c(n) E(0) = fGO(Rn+1) E(nf, o 0) dP. s

Here Pc- G3(Rn+') and r : Sn\Sn fl P1 -+ S" n P is given by np(y) = y'/iy'I, where y'

is the orthogonal projection of y onto P.

176

492

J. EELLS AND L. LEMAIRE

[Coron, Gulliver] have found the analogue of (12.29) for the 0-homogeneous extension of the Hopf map 0: aD2s - S' for n = 2, 4. Moreover [Brezis, Coron, Lieb 1, 2]: if yr: aD3 -* S2 is a non-constant map whose extension q5(x') = yr(x'/Ix'l) realises the minimum, then yi is the identity up to an isometry of S2. (12.30)

Minimising properties of harmonic diffeomorphisms of punctured

n-discs have been given by [Coron, Helein]. (12.31) The harmonic maps from a 3-dimensional domain M to S2 appear as models for the director of a liquid crystal with equal Frank constants. In this context, the liquid crystal is a fluid in a container M, containing rodlike molecules whose directions are specified by a unit vector field (the map from M to S2). These directions

are fixed along the boundary and the configuration assumes a position which minimises the energy (see [Eriksen] and [Almgren, Browder, Lieb]). Maps of surfaces (12.32) Recall (12.4) that any harmonic map 0: D2 -+ (N, h) which is constant on OD2 is constant on D2. In contrast, [Brezis, Coron] and [Jost 6] have shown that if (M, g) i s a compact s u r f a c e with boundary and f ' : : aM -+ (S2, h) is non-constant, then

there are at least two components of QM, S2) containing energy minimising harmonic maps (M, g) --* (S2, h).

(12.33)

In fact, lack of uniqueness is frequent for the Dirichlet problem, when no

restriction is imposed on the range. Here is an application due to [Ding 1] of Lusternik-Schnirelmann category, using the perturbation theory of [Uhlenbeck 2] and [Sacks, Uhlenbeck 1]. Let M be a compact Riemann surface with boundary and (N, h) a compact manifold. Set s = Inf{E(B): 0: S2 -+ (N, h) is a non-constant harmonic map};

= co if there are no such maps.

Suppose given a map yr: OM - N extendible to M, and let ba, = Inf {E(O): 0 e W,(M, N)}.

Take p > 2, so that the inclusion map .i. ,(M, N) " '2(M, N) is a homotopy equivalence. Let, be a component of (e,(M,N), and ,Yi.y, the component of Yi, y,(M, N) which contains it. If A is a subset of ,YJP, w, its category cats p (A) is the least integer k such that A can be covered by k closed subsets of _*'i w, each of which is contractible in Wi,W Set

Cf =Inf{SupA E: A c ;gyp and cat,, pW(A) W

j}.

Assume that we can find e > 0 such that for any p > 2, there is p E (2, p] with Ck, w < by, + s - e. Then there are at least k harmonic maps in YD

[Ding 1] has an analogous assertion if aM = .

(12.34) He draws the following consequence in case M is a planar domain. The general case requires only minor modifications.

177

ANOTHER REPORT ON HARMONIC MAPS

493

Let M be a compact Riemann surface with boundary, and yr : aM -+ S" a map which is non-constant on aM. Then yr has at least two harmonic extensions. (12.35)

In case M = D2, (12.34) was established by [Benci, Coron]. For

n > 3, their proof involves the existence (for p > 2) of non-contractible maps 0: S"-2 -+ 2,,,,(D2, S"), as well as a perturbation theorem based on ideas of [Sacks, Uhlenbeck 1]. Let g _ .i, y,(D2, S"), and assume for all a > 1 that there is a critical

point 0. of E, where E«(O) =

f

(1 + dq I2)° dx,

42

and that

lim E(0 J < Inf E(q) + 87t. all

oce

Then (0) is subconvergent to a harmonic map D2 _ S". [K.-C. Chang] has provided an approach to these ideas via the heat equation, in the spirit of (5.5). [Ji]. Any smoothly embedded circle in S" has at least two minimal coboundaries. The proof is based on the above mentioned result of [Benci, Coron], and uses a modified Lusternik-Schnirelmann category. (12.36) As for surfaces without boundary, we have [Jost, Schoen]: let (Mg) and (N, h) be two surfaces with boundary, and suppose aN convex. If there is a difeomorphism

yr: M -* N, then there is a harmonic d feomorphism 0 in'w(M, N), having least energy

amongst all djeomorphisms in ',,(M, N). [Coron, Helein] : 0 is in fact energy minimising in ',(M, N). Holomorphic maps Some properties of harmonic maps into manifolds with strongly negative curvature (4.10) extend to the boundary case in the following form. (12.37)

(12.38) Let (Mg) and (N, h) be two complex manifolds, and Mo a smooth domain in M with boundary aMo. A map q: Mo -+ N satisfies the tangential

Cauchy-Riemann equations on aMo if at each point x of aMo the differential d¢ restricted to the complex subspace T(3M0) f1 JT (aMo) is complex linear. If it denotes the projection of T(M) to that subspace, the condition can be written ab a¢ o7t = 0. Clearly, the restriction of a holomorphic map to aMo must satisfy this condition. (12.39) [Siu 5]. Let Mo be a smooth compact domain in a Kahler manifold (M, g) of complex dimension m > 2 with (m - l )-hyperconvex boundary aMo (4.32), which is strictly so somewhere. Let (N, h) be a Kahler manifold with very strongly negative curvature tensor (a)

If 0: (M0, g) - + (N, h) is harmonic, smooth up to the boundary aMo and satisfies

= 0 on aMo, then ¢ is holomorphic on Mo. (b) If V/: M. - (N, h) is a map smooth up to the boundary with ao yr = 0 on aMo, then there is a map 0: (Mo,g) --> (N, h), extending yi I.M, and holomorphic on M. The proofs of (a) and of (4.12) are closely related. That of (b) follows from (a) and Hamilton's existence theorem (12.6). A new ingredient is `Money's trick' to treat the boundary terms in the integrations by parts [Morrey 2]. ab

178

494

J. EELLS AND L. LEMAIRE

(12.40) [Siu 5] also provides several weaker conditions on the curvature to produce the same conclusions. For instance, we can take for (N, h) a quotient of an irreducible bounded symmetric domain of dime N >, 2, even though R" is not very strongly semi-negative. (12.41)

Here is a version [Shiga] of the classical theorem of Bochner and

Hartogs. Let M, N be two smooth bounded domains in a Stein manifold of complex dimension 2. If their boundaries are connected and CR-diffeomorphic (4.48), then they are biholomorphically equivalent.

(12.42) [Fefferman]. A biholomorphic map between two strictly pseudo-convex smooth domains in C" (that is, such that the Leviforms of their boundaries are definite) is smooth up to the boundary (n 3 2).

(12.43) [Nishikawa, Shiga]. Let (Mg) and (N, h) be Kahler manifolds of the same dimension > 2, and suppose that (N, h) has strongly negative curvature. Take compact smooth domains M, = M and No c N, and suppose 8Mo is (m-1)hyperconvex. If yr: Mo -> N, is a homotopy equivalence such that yi IIMO: 8Mo -* ON, is a CR-d feomorphism, then y/ I extends to a biholomorphic isomorphism M, -> No.

As in (4.13), the curvature hypothesis can be somewhat relaxed. Minimal maps (12.44) The distance between two continuous paths yro : to - N and V,: I, -> N on a manifold (N, h) is defined as the infimum over all homeomorphisms a: I, -+ to of max,, distN'(yr,(t), yro(a(t))). A Frechet curve is an equivalence class of paths at zero distance from each other. A sequence of such curves (C,) converges in the sense of Frechet to a curve C if and only if they admit parametrisations by paths converging uniformly to a parametrisation of C. A closed simple Frechet curve is called a Jordan curve. (12.45) Let M be a compact surface of type (p, k), that is, of genus p whose boundary consists of k circles Let (N, h) be a manifold without boundary, compact or satisfying the uniformity condition (11.24). Let 16 = be a set of k disjoint Jordan curves in N, such that there exist continuous Li-maps from M to N, mapping each B, monotonically on C, (that is, orientation preserving with inverse

image y/-'(y) connected for all y e Q. We say that two such maps 0, and 0, are relatively homotopic if and only if they are homotopic through a family of maps satisfying the same boundary conditions. (12.46) Let p denote a conformal structure on M. A conformal harmonic map 0: (M, p) - (N, h) is minimal. A map 0 which is an extremal of E for all variations of the conformal structure on M and all deformations of 0 in a relative homotopy class is conformal harmonic. These properties emerged from successive generalisations of a classical lemma of minimal surface theory [Douglas 1, 2, 3], [Morrey 1], [Sacks, Uhlenbeck 1, 2].

(12.47) [Heinz, Hildebrandt], [Meeks, Yau 3]. Let M be a smooth bounded planar

domain and 0: M -. (N, h) a conformal harmonic map such that for every oriented

179

ANOTHER REPORT ON HARMONIC MAPS

495

boundary component B1, 0 I B, is a monotonic representation of the oriented Jordan curve C,. Then O Id,,, is injective. (12.48) To prove the existence of a minimal map 0: M -* (N, h) in a given class of maps, we need an irreducibility condition of the type introduced by [Douglas 2, 3] and [Morrey 1]. The present form can be found in [Jost 9].

Denote by .B the class of continuous L'-maps from M to N, mapping each B, monotonically on C, and inducing the conjugacy class of homomorphisms 0: n,(M) -* rJN) First, define a primary reduction M' of M as a surface obtained from either of the following operations. (a)

Take an essential Jordan curve f in IntM whose images by the maps in

.r,, are contractible in N. Cut M along ,6 and collapse each of the two resulting curves to a point. (b) Take an arc 7 in M joining two points of some B, which is not homotopic to an arc of Bi through arcs with end-points on B, whose X8-images are homotopic to arcs in C,. Cut M along rl; the two parts of B, and the two copies of ri form two new boundary curves. Collapse one of them to a point. Now define E(p, le, X-) = inf (lim inf E(cb8, p,)) for all sequences of conformal structures p, on M and continuous L'-maps q,: (M, p,) --> (N, h) inducing 0 on the fundamental groups and mapping each B, monotonically on a curve C,., in N, where (C,.,) converges to C, in the sense of Frechet (1 < i < k). Define E*(p,W, .*') = co if p = 0 and k = 1, or if there are no primary reductions of M; and E*(p, cf, A 'O) = inf E(p', ', B) otherwise, where the infimum is taken over all surfaces homeomorphic to primary reductions of M. (12.49)

The irreducibility condition is E(p,',.*) < E*(p,(,.re).

(12.50) The following solution of the Plateau problem emerged from successive generalisations of the work of [Douglas 1, 2, 3] and [Rado] by [Morrey 1], [Sacks, Uhlenbeck 2], [Schoen, Yau 3], [Lemaire 5] and [Jost 9]. Let M be a compact surface of type (p, k) and (N, h) a manifold satisfying the

uniformity condition (11.24). Let IV = (C) be a family of Jordan curves in N and .aCoe a class as above, satisfying the irreducibility condition (12.49). Then there is a conformal structure on M and a conformal harmonic map 0: (M, p) --> (N, h) in ,Y,. Furthermore

E(0) = E(p, 9,-f.) and 0 minimises area. Finally, if n2(N) = 0, a variation of the method yields a map ¢ relatively homotopic to a given map W in X0.

We remark that the irreducibility condition is automatically satisfied if the homomorphism 0 is injective. (12.51) The proof of (12.50) requires two minimising processes. First find harmonic maps for given conformal structures on M and Dirichlet data (using

(12.8)); then minimise the energy amongst the conformal structures and the Dirichlet

data corresponding to the given Jordan curves. Some irreducibility hypothesis is needed in the second step because the Teichmuller space of conformal structures on M is not compact. However, the boundary of a suitable compactification consists of primary reductions of M, and (12.49) ensures precisely that they will not appear in a minimising process.

180

J. EELLS AND L. LEMAIRE

496

(12.52) A classical illustration of the need for some such hypothesis is based on the fact that two circles placed far from each other in 183 do not span a minimal cylinder; indeed, in a minimising sequence of conformal structures on the cylinder it gets thinner and thinner (conformally: longer and longer) and its limit is two discs spanning the circles : a minimising primary reduction.

(12.53) We now introduce a coercivity condition. Let M be a surface of type (p,k) and 9 = (C), ,,k a set of k disjoint Jordan curves in (N,h). Suppose that for every conformal structure u on M and for any map w : 8M --+ (N, h) such that yr(B1) = C,, the Dirichlet problem has a non-degenerate solution in each component of W,,,(M, N) (in the sense that VdE is non-degenerate), and that if c 0,: (M, p) (N, h) are two harmonic maps relatively homotopic in the sense of (12.45), then

sup d''(Oo(x), 0,(x)) < Csup d'"(co(x), 0,(x)) M

OM

(12.54) This condition implies in particular that for a given y,, the solution of the Dirichlet problem is unique in each component of W,(M, N). It is satisfied if Riem' < 0, or if all maps under consideration have images in a geodesically small disc (3.14). (12.55) The next result is due to [Chang, Eells 1, 2]. It is a generalisation of a theorem of [Morse, Tompkins 1, 2, 3] and [Shiffman 1, 2, 3], who treated the case

p = 0 and (N, h) = 18". That case was reproved by [Struwe 2] for M = D2, (N, h) = 18";

and his method was then extended to the present situation. Let Y be a relative homotopy class of maps of M to (N, h), and suppose that (N, h) satisfies the uniformity condition (11.24), and 9 and .*' the irreducibility condition (12.49), where .B is the class of maps inducing the same homomorphism 0 on fundamental groups as the maps in .°, and the coercivity condition (12.53).

Suppose further that any two E-minima can be joined by a path (y) such that E(y,) < E*(p, `', °e) If 0o : (M, µo) -> (N, h) and 01 : (M, ,u1) -- (N, h) are relatively homotopic isolated minimal maps, carrying each B1 to C, monotonically, then there is a conformal structure p on M and a conformal harmonic map 0: (M, p) --> (N, h), carrying each B1to C, monotonically, which is relatively homotopic to 0, and Y' and is not an Eminimum. (12.56) Here are the main ideas of the proof. Non-degeneracy of VdE implies smooth dependence on the Dirichlet data (2.20), which can be expressed in terms of

the Li12 topology on the space of maps 3M-> N. Together with Mumford's compactness theorem on the space of moduli of M, this yields the compactness condition of Palais and Smale for E; and hence a mountain pass theorem.

When all the critical points of E are isolated, we also have versions of the Lusternik-Schnirelmann category theorem and of the Morse inequalities [Chang, Eells 2]. (12.57) The problem of minimising the volume in higher dimensions is much less well understood. However, there is an important development due to [White 1]. If M

is a compact m-manifold with boundary with 3 < m < 6 and yr: 8M-> 18'"+' an embedding, then there is a Lipschitz map 0: M-- 18'"+' extending yi and minimising volume.

His proof is based on geometric measure theory.

181

ANOTHER REPORT ON HARMONIC MAPS

497

Minimal embeddings (12.58)

In general, a minimal map need not be an embedding or even an

immersion. In particular, it could have branch points. However, if dim N = 3, an area minimising map cannot have any interior branch point (5.18). (12.59) The following embedding theorem is due to [Meeks, Yau 3]. Let (N, h) be a compact 3-manifold with convex boundary and C a Jordan curve embedded in ON. If C is contractible in N, then any area minimising map 0: D2 -a (N, h) such that 0 10D' maps 8D2 monotonically onto C is an embedding. Furthermore, if ¢a and 01 are two

such maps, then either they coincide up to conformal reparametrisation of D2 or co(D2) (1 c1(D2) = C. If C is smooth, then so is 0. Here and in (12.61) below, the existence of a minimal embedding is also due to [Almgren, Simon] in case N c 683 The proof of Meeks and Yau involves an approximation argument to reduce the problem to the real analytic case, the tower construction in topology, and a surgery argument to show that a minimising map cannot have unnecessary folds. These tools are used to alter a non-embedding to one of smaller area. (12.60) Every compact 3-manifold N carries a Riemannian structure h with respect to which ON is convex. Consequently, (12.59) implies Dehn's lemma of topology. If C is a Jordan curve in ON which is contractible in N, then C bounds an

embedded disc in N.

(12.61) [Meeks, Yau 2]. Let (N, h) be an orientable 3-manifold with convex a family of disjoint Jordan curves in ON such that there is boundary, and W = (C)1 a map of a possibly disconnected plane domain Mo to N, carrying OM, onto W. Such a domain has at most k components. Let V(') be the infimum of the areas of such maps of domains with r components. If V,(le) < Vr+1(W), then there exists a conformal harmonic immersion ¢ of a planar domain with r components in N, spanning W, with smallest possible area V,(,W). Furthermore, any such map is an embedding. (12.62) [Meeks, Yau 4] prove the following. Let (N, h) be an analytic compact 3-manifold such that ON has non-negative mean curvature; and No c ON a compact connected surface with boundary 8Na consisting of rectifiable Jordan curves, such that iv1(No) injects into tv1(N). Then there is a Riemann surface M with boundary aM homeomorphic to aNa and a minimal embedding 0: M -i (N, h) which is areaminimising, and such that 0lane is a homeomorphism onto ONo. [Hardt, Simon]. Any finite union of disjoint Jordan curves in 683 spans a compact embedded minimal surface.

[Meeks, Yau 2] have produced equivariant versions of Dehn's lemma (12.59), (12.61), the loop theorem (12.72) below, and the sphere theorem (5.35). These have contributed much to our understanding of the action of finite groups G of orientation-preserving isometries of 3-manifolds (N, h), basically because a minimal surface in (N, h) behaves well under such an action. In broad terms: G-action on a compact (1) The equivariant sphere theorem ensures that a manifold must split-so that in analysing it, we can suppose N prime (that is, not expressible as a connected sum of two manifolds, neither of which is homeomorphic to S3) [Meeks, Yau 1]. (12.63)

182

498

J. EELLS AND L. LEMAIRE

(2)

The equivariant loop theorem, together with a theorem of Thurston on

hyperbolic structures and Bass's analysis of finitely generated subgroups of GL (C2), are key ingredients in the proof of P. A. Smith's conjecture that a finite cyclic group of orientation-preserving diffeomorphisms of S3 has fixed point set either void or an unknotted circle (that is, bounding a disc); equivalently, that action is conjugate to a linear action [Bass, Morgan]. (3)

A covering space of an orientable irreducible 3-manifold is irreducible

[Meeks, Simon, Yau]. (4) Any periodic diffeomorphism of l3 is conjugate in the diffeomorphism group D(ff83) to a rotation [Meeks, Yau 1]. (5) Any compact group G acting smoothly on Ifl is isomorphic to a subgroup of

SO(3), and the action is conjugate to an orthogonal action ([Meeks, Yau 5], and Thurston for the exceptional case G = A5). (12.64) Other embedding theorems for minimal surfaces with boundary were obtained by [Freedman, Hass, Scott] and [Jost 13], in classes of maps in which it is assumed a priori that embeddings exist (which is not the case in (12.59)). We shall state them in a more general setting in (12.73) below.

Free boundary problems (12.65)

Various free boundary problems can be posed as follows. Let M be a

manifold with boundary OM and N, a closed submanifold of N. For a fixed homotopy class Y of maps from 3M to No, is there a harmonic map 0: (M, g) -+ (N, h) such that O(OM) c No, l3 0l3M is perpendicular to N, and 0 lad, C- Y? Can we prescribe the relative

homotopy class of 0? If every map from OM to N. is homotopic to a constant, is there a non-constant solution 0? Is there an embedded solution? The condition that the normal derivative 0, 0 be perpendicular to N, is naturally associated to the variational problem under consideration : indeed, it is automatically satisfied by an extremal of E amongst maps 0 such that O(5M) c No.

(12.66) We can also consider partially free problems, for which we prescribe Dirichlet data on part of OM and free data on its complement. From the point of view of existence, those are easier because the Dirichlet data on part of the boundary will often help the convergence of a minimising sequence. (12.67) Regularity theory near the boundary of a free boundary problem is more difficult than for the Dirichlet or Neumann problem. It has been developed through the efforts of [Baldes 1], [Dziuk 1, 2], [Griiter, Hildebrandt, Nitsche], [Gulliver, Jost], [Hamilton 1], [Jager], [Jost 14], [Lewy 1], [Ye 1, 2]. Usually the geometry of No plays an important role. The following two results concern the case where N, is totally

geodesic. (12.68) Let (M, g) and (N, h) be compact manifolds with boundary, and suppose that Riem' < 0 and ON is totally geodesic. Then the free boundary problem z(c) = 0,

0(3M) c ON and a.0 laM 1 ON has a solution in every relative homotopy class [Hamilton 1].

183

ANOTHER REPORT ON HARMONIC MAPS

499

(12.69) Let (M,g) be a compact manifold and No a totally geodesic submanifold of the manifold (N, h). Suppose that D(yo, p) is a geodesically small disc of (N, h) for every yo E No, which is a normal chart for both N and No. Let Mo be open in 8M, and put M, = M\Mo. Suppose that M, has positive measure in M. Let V: (M, g) - (N, h) be a map such that V/(M) c D(yo, p) and yi(M0) c No. Then there is a minimising harmonic map 0: (M, g) -* (N, h) such that M = yr I M and O(M,) Na [Baldes 1].

If w is C', then ¢ is continuous on M and smooth in its interior, and as regular on M,, and M, as the boundary data permit. (12.70) When N, is not totally geodesic, [Gulliver, Jost] have given an example of a weak solution to the free boundary problem which is not continuous. They have

also given a condition on the oscillation of a weak solution near the boundary ensuring its continuity. (12.71) As a special case, they consider the situation where (M, p) is a Riemann surface of type (p, k) and No a union of k Jordan curves in N. If 7r2(N) = 0 and N,

satisfies an irreducibility condition similar to (12.49) (but with fixed complex structure,u) they prove the existence of a solution of the free boundary problem in every

relative homotopy class, minimising E and mapping the boundary components monotonically to the Jordan curves.

Note that the boundary data are exactly those of the Plateau problem (12.50); however, the conformal structure ,u is fixed here, so that the solution is harmonic but not necessarily minimal. (12.72) The following theorem of [Meeks, Yau 1]-in the spirit of their version of the Dehn lemma (12.59)-provides an analytic proof of the general loop theorem in topology. Let N be a compact 3-manifold with convex boundary, and N, the disjoint union of some components of ON. Let K be the kernel of i,: 7t,(N0) -p n,(N), where i is the

inclusion N, -+ N. Then: (a) there is a finite number of conformal harmonic maps 0i ... yb : (D2, 8D2) -> (N, No) such that

(i) 0, has area minimal amongst all maps 0 from D2 to N whose boundary ¢(aD2) represents a non-trivial element in K, (ii) for each i, 0, has area minimal amongst all maps 0 such that O(8D2) does not belong to the smallest normal subgroup of tr,(N0) containing [¢,(8D2), ..., t_,(aD2)], (iii) the discs 0,(D2) are orthogonal to N, along their boundaries, (iv) K is the smallest normal subgroup of rc,(S) containing all {[0,(cD2)]} (1 < i 5 k),(b)

any set of maps (0,. .. 0,j satisfying (i) and (ii) are embeddings and have

mutually disjoint images; (c) any other maps satisfying (i) and (ii) are either a reparametrisation of 0, or have images disjoint from theirs.

We stress that it is not assumed a priori that the elements of K contain curves bounding an embedded disc.

184

500

J. EELLS AND L. LEMAIRE

For surfaces of type (p, k), the following solution to the embedding problem with partially free boundary data was given (in a more general form) by [Jost 13] (with previous contributions by [Freedman, Hass, Scott]). (12.73)

Let (N, h) be a compact three dimensional manifold whose boundary ON has non-

negative mean curvature. Let Kc N be a closed subset with sufficiently regular boundary, and let (Cl)t_1.....1 be disjoint Jordan curves in 8N\K. Consider the space ..((p, 1W, 8K) of oriented embedded surfaces of genus p in N, such that l

k

aM=UClU U y1, i-1

1-1+1

with yJ Jordan curves in 8K. Set V(p, ', 8K) = inf {Area (M) I Me.df(p, le, 8K)}.

Call V*(p, , 3K) the infimum of the area on all possibly disconnected surfaces satisfying the same boundary data. If V(p, IF, 8K) < V*(p,16, 8K), then there exists an embedded minimal surface in N\K of genus p, having le = (Cr)t_1...l as fixed boundary curves and possibly some free boundary curves in K. If, furthermore, K has non-positive mean curvature, then the number k-I of free boundary curves can be prescribed as well. In this statement, the existence of an embedding of M in N satisfying the boundary conditions is assumed.

The proof relies on geometric measure theory. Of course, the condition V(p, ', 8K) < V *(p, ', 8K) is similar in spirit to the irreducibility condition (12.49). (12.74) In the free boundary problem for minimal surfaces, it can happen that the infimum of the area in a homotopy class is zero. However, various minimax methods and those involving Lusternik-Schnirelmann category are available to give the following results, in which saddle solutions appear.

(12.75) [Struwe 1]. If No = R3 is a closed surface of genus zero, there is a conformal harmonic map 0: (D2, OD2) --* (683, No) with 8, 01 No along 3D2. [Griiter, Jost]. If No is the boundary of a strictly convex body B c 683, then there is an embedded minimal disc in B meeting 8B orthogonally.

For any closed surface Na c 683 [Tolksdorf 3] has shown how to decompose any non-trivial free homotopy class of loops 3D2 --. No into finitely many such classes, each of which is associated to a conformal harmonic map (12.76)

(D2, 7D2) --* (683, No) solving the free boundary problem.

(12.77) [Jost 13]. Let B be a 3-disc in a 3-manifold (N, h) whose boundary 8B has non-negative mean curvature. Then B contains an embedded minimal 2-sphere or an embedded minimal 2-disc meeting 8B orthogonally.

Dirichlet problem for minimal graphs

Given a map w : M -+ N, find a map 0: M - N with 0 I aM = W I , M and whose graph map D: (M, g+ qS*h) -+ (M x N, g x h) is harmonic. (Equivalently (11.76), 4)2 is harmonic.) (12.78)

185

501

ANOTHER REPORT ON HARMONIC MAPS

That problem has much greater diversity than (12.2), even if we restrict ourselves to Euclidean domains. (12.79) Let U c (lm, go) be a bounded smooth domain and yr : U --+ (W , ho) a map, where g, and ha denote Euclidean metrics. First of all, we require that U be

strictly convex, for without some hypothesis of that kind, we cannot expect existence [Finn]. Then for n = 1, the problem has a unique solution [Jenkins, Serrin]; and the graph of the solution minimises volume.

For m = 2 and n arbitrary, there is a solution to (12.78); it may not be unique. There can be V-unstable solutions, as given by [Lawson, Osserman]. For m >, 4, there may not be even a Lipschitz solution [Lawson, Osserman].

ITEMS ADDED IN PROOF.

(4.45) If (M, g) is a compact Kahler manifold, its Ricci curvature determines a closed (1, 1)-form pM whose cohomology class is the first Chern class: [wM ]m-1

f

dx = c1(M) U

M

(m-1)

[M].

If that is positive, we shall write c1(M) > 0. Let (M, g) be a compact Kahler manifold, and 0: (M, g) -+ (N, h) a (1, 1)-geodesic map to a compact Riemannian surface. Then 0 is ± holomorphic, provided

(a) c1(M) > 0 [Ohnita, Udagawa 2]; or 0 and [cb*w"] = a[wM] for some ac- R [Naito 2].

(b)

The case dime M = 1 is due to [Eells, Wood 1]. (6.45) [Ohnita, Udagawa 2] provide many situations in which (1, 1)-geodesicity implies drastic restrictions. For instance, let (M,g) be a compact Kahler manifold with c1(M) > 0 (as in (4.45)) and fi2(M) = 1. Then (a) if (N, h) is a manifold with 9'1,, > 0 and 0: (M, g) -+ (N, h) a non-constant (1, 1) - geodesic map, then dime M = 1, and 0 is a minimal branched immersion ; (b) if N is an irreducible Hermitian symmetric space of compact type, then any stable (1, 1)-geodesic map 0: (M, g) -+ N is ±holomorphic. (10.40)

[Eells, Ratto] have just completed a study of equivariant harmonic maps

between ellipsoids-in the spirit of (10.25){10.39), but requiring both direct variational methods and Morse theory. Here are samples of their results: We are concerned with Euclidean ellipsoids of the form Qm-1(a, b) = {(x, y) E RP x R":

IXI2/a2 + I y12/b2 = 1),

where p, r > 2, a, b > 0 and p + r = m. Its points are represented by z = a sins . x + b cos s. y for x e SP-1, y e Sr-1 and 0 S s < 2ir. The induced Riemannian metric on Qm'1(a, b) is

g = a2 sin2S.g,sp-1+b2 cos2s.gs +(b2 sin' s+a2 cos2s)ds2.

186

502

J. EELLS AND L. LEMAIRE

Take ux : S' -> S1 as in (10.32), and v : Ss-2 --- Si-2 the identity

(a) ExAMPLE.

map (n >, 3). Then for any a map of degree k, of the form

b > 0 there are c, d > 0 and an equivariant harmonic

9 = u*a v - Q'(a, b) Q'(c, d). (b) For any eigenmaps u : SP-1 , Sq-1 and v : Sr-1 --* Ss-1 and a, b > 0, there is an equivariant harmonic map 0 = u*. v : QP+r-1(A, b) -+ S°+8-1

homotopic to u*v.

(c) With the same hypotheses, there are c, d > 0 and an equivariant harmonic

/

map

cz = u*av: (d) ExAMPLE.

SP+r-1

- Q4+-1(c, d).

If (n - 3)2/4(n - 2) < d 2/c2, then every homotopy class in 7c (Sn)

has an equivariant harmonic representative Q'(c, d) - Q"(c, d).

(e) For any integers k, l there is an equivariant harmonic map 0,,,: Q3(a, b) -* S2 with Hopf linking invariant kl if and only if b/a = Il/kI. Furthermore, 0 t is a harmonic morphism.

References This bibliography contains only papers referred to in the text. U. ABRESCH

1. 'Constant mean curvature tori in terms of elliptic functions', J. Reine Angew. Math. 374 (1987) 169-192 T. ADAM and T. SUNADA

1. 'Energy spectrum of certain harmonic mappings', Compositio Math. 56 (1985) 153-170

2. `Geometric bounds for the number of certain harmonic mappings', Differential geometry of submanifolds, Lecture Notes in Math. 1090 (Springer, Berlin, 1984), pp. 24-36. J. F. ADAMS

1. 'On the groups J(X)', Topology 3 (1965) 137-171; 5 (1966) 21-71. 2. 'Maps from a surface to the projective plane', Bull. London Math. Soc. 14 (1982) 533-534. L. V. AHLFORS

1. 'Curvature properties of Teichmuller's spaces', J. Analyse Math. 9 (1961) 161-176. A. R. AITHAL

1. 'Harmonic maps from S2 to HP2', Osaka J. Math. 23 (1986) 255-270, 2. 'Harmonic maps from S2 to G2(C5)', J. London Math. Soc. 32 (1985) 572-576. A. D. ALEXANDROV

1. 'Uniqueness theorems for surfaces in the large, V', Amer. Math. Soc. Transl. 21 (1962) 412-416. F. J. ALMGREN, W. BROWDER and E. H. LIEB

1. 'Co-area, liquid crystals, and minimal surfaces', preprint, Princeton University, 1987. F. J. ALMGREN and E. H. LIEB

1. 'Singularities of energy-minimising maps from the ball to the sphere', Bull. Amer. Math. Soc. 17 (1987) 304-306. F. J. ALMGREN and L. SIMON

1. 'Existence of embedded solutions of Plateau's problem', Ann. Scuola Norm. Sup. Pisa 6 (1979) 447-495. H. W. ALT

1. 'Verzweigungspunkte von H-Flachen I', Math. Z. 127 (1972) 333-362. 2. 'Verzweigungspunkte von H-Flachen II', Math. Ann. 201 (1973) 33-55. M. T. ANDERSON

1. 'The Dirichlet problem at infinity for manifolds of negative curvature', J. Di( 701-721.

Geom. 18 (1983)

187

ANOTHER REPORT ON HARMONIC MAPS

503

M. T. ANDERSON and R. SCHOEN

1. 'Positive harmonic functions on complete manifolds of negative curvature', Ann. of Math. 121 (1985) 429-461. A. ANDREOTTI and E. VESENTINI 1.

' Carleman estimates for the Laplace-Beltrami equation on complex manifolds', Inst. Hautes Etudes Sci. Publ. Math 25 (1965) 81-130.

M. F. ATIYAH

1. 'Instantons in two and four dimensions', Comm. Math. Phys. 93 (1984) 437-451. M. F. ATYAH, N. J. HITCHIN and I. M. SINGER

1. 'Self-duality in four-dimensional Riemannian geometry', Proc. Roy. Soc. London A362 (1978) 425-461. P. AVILES, H. I. CHOI and M. MICALLEF

1. 'The Dirichlet problem and Fatou's theorem for harmonic maps', preprint, 1987. C. BAIKoussls and T. KouFoGIORGOs

1. 'Harmonic maps into a cone', Arch. Math. (Basel) 40 (1983) 372-376. P. BAIRD

1. Harmonic maps with symmetry, harmonic morphisms and deformations of metrics, Research Notes in Math. 87 (Pitman, 1983). 2. 'Harmonic morphisms onto Riemann surfaces and generalized analytic functions', Ann. Inst. Fourier 37 (1987) 135-173. P. BAIRD and J. EELLS

1. 'A conservation law for harmonic maps', Geometry Symp. Utrecht 1980, Lecture Notes in Math. 894 (Springer, Berlin, 1981), pp. 1-25. P. BAIRD and J. C. WooD

1. 'Bernstein theorems for harmonic morphisms from R3 and S3', Math. Ann. 280 (1988) 579-603. A. BALDES

1. 'Harmonic mappings with partially free boundary', Manuscripta Math. 40 (1982) 255-275. 2. 'Stability and uniqueness properties of the equator map from a ball into an ellipsoid', Math. Z. 185 (1984) 505-516. S. BANDO

1. 'On the classification of three dimensional compact Kaehler manifolds of nonnegative bisectional curvature', J. Di(. Geom. 19 (1984) 283-297. S. BANDO and Y. OHNITA

1. 'Minimal 2-spheres with constant curvature in P .(C)', J. Math. Soc. Japan 39 (1987) 477-487. D. BARBASCH, J. F. GLAZEBROOK and G. TOTH

1. 'Harmonic maps of complex projective spaces to quotients of spheres', preprint, Rutgers University, 1987.

J. L. BARBOSA and M. Do CARMO

1. 'Stability of hypersurfaces with constant mean curvature', Math. Z. 185 (1984) 339-353. J. L. BARBOSA, K. KENMOTSU and G. OSHIKIRI

1. 'Foliations by hypersurfaces with constant mean curvature', preprint. P. DE BARTOLOMEIS

1. Sur I'analyticite complexe de certaines applications harmoniques', C.R. Acad. Sci. Paris A 284 (1982) 525-527. H. BASS and J. W. Morgan (editors) 1. The Smith conjecture. (Academic Press, 1984). V. BENCI and J.-M. CORON

1. 'The Dirichlet problem for harmonic maps from the disk into the Euclidean n-sphere', Ann. Inst. H. Poincare Anal. Non Lin. 2 (1985) 119-141. F. BEniuEL and X. ZHENG

1. 'Sur la densite des fonctions regulieres entre deux varietes dans des espaces de Sobolev', C. R. Acad. Sci. Paris A303 (1986) 447-449. B. BomARSKI and T. IwANIEc

1. 'p-harmonic equation and quasilinear mappings', preprint, Bonn, 1983. J. BOLTON, G. R. JENSEN, M. RIGOLI and L. M. WOODwARD

1. 'On conformal minimal immersions of St in CP"', preprint. E. BoMBmu, E. DE GIORGI and E. Glusn

1. 'Minimal cones and the Bernstein problem', Invent. Math. 7 (1969) 243-268.

188

J. EELLS AND L. LEMAIRE

504 H. J. BORCHERS and W. D. GARBER

1. 'Local theory of solutions for the 0(2k + 1) c-model', Comm. Math. Phys. 72 (1980) 77-102. 2. 'Analyticity of solutions of the O(N) non-linear a-model', Comm Math. Phys. 71 (1980) 299-309 A. BOREL and F. HIRZEBRUCH

1. 'Characteristic classes and homogeneous spaces 1', Amer. J. Math. 80 (1958) 458-538. H. BREzis and J.-M. CORON

1. 'Large solutions for harmonic maps in two dimensions', Comm. Math. Phys. 92 (1983) 203-215. H. BREZIS, J.-M CORON and E. H LIES

1. 'Estimations d'energie pour les applications de R3 a valeurs dans S2', C.R. Acad Sci. Paris 303 (1986) 207-210. 2. 'Harmonic maps with defects', Comm Math. Phys. 107 (1986) 649-705. R.BRYANT

1. 'Submanifolds and special structures on the octonians', J. Diff. Geom. 17 (1982) 185-232. 2. 'Conformal and minimal immersions of compact surfaces into the 4-sphere', J. Diff. Geom. 17 (1982) 455 473. 3. 'A duality theorem for Willmore surfaces', J. Diff. Geom. 20 (1984) 23-53. 4. 'Lie groups and twistor spaces', Duke Math. J. 52 (1985) 223-261. 5. 'Minimal surfaces of constant curvature in S", Trans. Amer. Math. Soc. 290 (1985) 259-271 G. BUNTING

1. 'Proof of the uniqueness conjecture for black holes', thesis, University of New England, 1983 D. BURNS

1. 'Harmonic maps from CP' to CP"', Proc. Tulane Conf., Lecture Notes in Math. 949 (Springer, Berlin, 1982), pp. 48-56. D. BURNS and P. DE BARTOLOMEIS

1. 'Applications harmoniques stables dans P"', Ann. Sci. Ecole Norm. Sup. (4) 22 (1988) 159-177. D. BURNS, F. BURSTALL, P. DE BARTOLOMEIS and J. RAWNSLEY

1. 'Stability of harmonic maps of Kahler manifolds', preprint, University of Warwick, 1987. K. BURNS

1. 'Convex supporting domains on surfaces'. Bull. London Math. Soc. 17 (1985) 271-274. F. E. BURSTALL

1. 'Harmonic maps of finite energy from non-compact manifolds', J. London Math. Soc. 30 (1984) 361-370.

2. 'Non-linear functional analysis and harmonic maps', thesis, University of Warwick, 1984. 3. 'Twistor fibrations of flag manifolds and harmonic maps of a 2-sphere into a Grassmannian', Proc. Santiago 1984, Research Notes in Math. 131 (Pitman, 1985), pp. 7-16. 4. 'A twistor description of harmonic maps of a 2-sphere into a Grassmannian', Math. Ann. 274 (1986) 61-74. F. BURSTALL and J. RAWNSLEY

1. 'Spheres harmoniques dans les groupes de Lie compacts et courbes holomorphes dans les espaces homogenes', C.R. Acad. Sci. Paris A 302 (1986) 709-712. 2. 'Stability of classical solutions of two-dimensional Grassmannian models', Comm. Math. Phys. 110 (1987) 311-316.

3. 'Twistor theory for Riemannian symmetric spaces', Sem. Bianchi, Springer Lecture Notes, to appear. F. BURSTALL, J. RAWNSLEY and S. SALAMON

1. 'Stable harmonic 2-spheres in symmetric spaces', Bull. Amer. Math. Soc. 16 (1987) 274-278. F BURSTALL and S. SALAMON

1. 'Tournaments, flags and harmonic maps', Math. Ann. 277 (1987) 249-265. F. BURSTALL and J. C. WOOD

1. 'The construction of harmonic maps into complex Grassmannians', J. Diff. Geom. 23 (1986) 255-297. E. CALABI

1. 'Minimal immersions of surfaces in Euclidean spheres', J. Diff. Geom. 1 (1967) 111-125. 2. 'Quelques applications de ]'analyse complexe aux surfaces d'aire minima', Topics in Complex Manifolds (Universite.de Montreal, 1967), pp. 59-81.

3. 'An intrinsic characterization of harmonic one-forms', Global analysis (Princeton University Press, 1969), pp. 101-117. H: D. CAO and B. CHOW

1. 'Compact Kahler manifolds with nonnegative curvature operator', Invent. Math. 83 (1986) 553-556.

189

ANOTHER REPORT ON HARMONIC MAPS

505

J. CARLSON and D. TOLEDO

1. 'Harmonic mappings of Kahler manifolds to locally symmetric spaces', preprint, University of Utah, 1987.

M DO CARMO and H. B. LAWSON

1. 'On Alexandrov-Bernstein theorems in hyperbolic space', Duke Math. J. 50 (1983) 995-1003. M. DO CARMO and C. K. PENG

1. 'Stable complete minimal surfaces in 18' are planes', Bull. Amer. Math. Soc. 1 (1979) 903-906. S. CARTER and A WEST

1. 'A characterisation of isoparametric hypersurfaces in spheres', J. London Math. Soc. 26 (1982) 183-192. T. E. CECIL and P. J. RYAN

1. 'Tight spherical embeddings'. Lecture Notes in Math. 838 (Springer, Berlin, 1981), pp. 94-104. K.-C. CHANG

1. 'Heat flow and boundary value problem for harmonic maps', prepnnt, Courant Institute, 1988. K.-C. CHANG and J. EELLS

1. 'Harmonic maps and minimal surface coboundaries', Lefschetz Centenary, Mexico (1984), Contemp. Math. 58 III (1987), pp. 11-18. 2. 'Unstable minimal surface coboundaries', Acta Math. Sinica (N.S.) 2 (1986) 233-247. B: Y. CHEN, J: M. MORVAN and T. NoRE

1. 'Energy, tension and finite type maps', Kodai Math. J. 9 (1986) 406-418. B.-Y. CHEN and T. NAGANO

1. 'Totally geodesic submanifolds of symmetric spaces 1', Duke Math. J. 44 (1977) 745-755. 2. 'Totally geodesic submanifolds of symmetric spaces II', Duke Math. J. 45 (1978) 405-425. W.-H. CHIN

1. 'The geometry of Grassmann manifolds as submanifolds', Adv. in Math. (China) 16 (1987) 334-335. X.-P. CHEN

1. 'Harmonic mapping and Gauss mapping', Proc. 1981 Shanghai-Hefei Sympos. Diff. Geom. Diff. Eq. (Sci. Press, Beijing, 1984), pp. 51-53. S. Y. CHENG

1. 'A characterization of the 2-sphere by eigenfunctions', Proc. Amer. Math. Soc. 55 (1976) 379-381. 2. 'Liouville theorem for harmonic maps', Proc. Sympos. Pure Math. 36 (1980) 147-151. S. S. CHERN

1. 'On the curvature of a piece of hypersurface in Euclidean space', Abh. Math. Sem. Hamburg 29 (1965) 77-91. 2. 'Minimal surfaces in Euclidean space of N dimensions', Sympos. in Honor of Marston Morse (Princeton University Press, 1965), pp. 187-198. S. S. CHERN, M. DO CARMO and S. KOBAYASIH

1. 'Minimal submanifolds of a sphere with second fundamental form of constant length', Proc. Con!. for M. Stone (Springer, New York, 1970), pp. 59-75. S. S. CHERN and R. OSSERMAN

1. 'Complete minimal surfaces in Euclidean n-space', J. Analyse Math. 19 (1967) 15-34. S. S CHERN and E. SPANIER

1. 'A theorem on orientable surfaces in four-dimensional space', Comment. Math. He/v. 25 (1951), 205-209. S. S. CHERN and J. G. WOLFSON

1. 'Minimal surfaces by moving frames', Amer. J. Math. 105 (1983), 59-83. 2. 'Harmonic maps of S2 into a complex Grassmann manifold', Proc. Nat. Acad. Sci USA 82 (1985) 2217-2219.

3. 'Harmonic maps of the two-sphere into a complex Grassmann manifold II', Ann. of Math. 125 (1987) 301-335. H. 1. CHOI

1. 'On the Liouville theorem for harmonic maps', Proc Amer. Math. Soc. 85 (1982) 91-94. 2. 'Asymptotic Dirichlet problems for harmonic functions on Riemannian manifolds', Trans. Amer. Math. Soc. 281 (1984) 691-716 H. I. CHOI and R. SCHOEN

1. 'The space of minimal embeddings of a surface into a three-dimensional manifold of positive Ricci curvature', Invent. Math. 81 (1985) 387-394.

190

506

J. EELLS AND L. LEMAIRE

Y.CHOQUET-BRUHAT

1. 'Global existence theorems for hyperbolic harmonic maps', Ann. Inst. Poincare Phys. Theor. 46 (1987) 97-111. J.-M. CORON and R. GULLIVER

1. 'Minimizing p-harmonic maps into spheres', preprint, Max-Planck Institute, Bonn, 1987. J.-M. CottON and F. HELEIN

1. 'Harmonic diffeomorphisms, minimizing harmonic maps and rotational symmetry', preprint, Ecole Polytechnique, Paris, 1988. C. COSTA

1. `Example of a complete minimal immersion in R3 of genus one and three embedded ends', Bol. Soc. Brasil Math. 15 (1984) 47-54. C. B. CROKE

1. 'Lower bounds on the energy of maps', Duke Math. J. 55 (1987) 901-908. M. DA czER and D. GROMOLL

1. 'Real Kaehler submanifolds and uniqueness of the Gauss map', J. Diif. Geom. 22 (1985) 13-28. M. DAIczER and L. RODRIGUEZ

1. 'Rigidity of real Kaehler submamfolds', Duke Math. J. 53 (1986) 211-220. M. DAjczER and G. THORBERGSSON

1. 'Holomorphicity of minimal submanifolds in complex space forms', Math. Ann. 277 (1987) 353-360. J. DAVIDOV and 0. MUSKAROV

1. 'Existence of holomorphic functions on twistor spaces', Bull. Soc. Math. Belgique B40 (1988). D. DETURCK and J. KAZDAN

1. 'Some regularity theorems in Riemannian geometry'. Ann. Sci. Ecole Norm. Sup. 14 (1981) 249-260. A. M. DIN and W. J. ZAKRzewsKu

1. 'General classical solutions in the CP"-' model', Nucl. Phys. B. 174 (1980) 397-406. 2. 'Properties of the general classical CP'-' model', Phys. Lett. 95B (1980) 419-422. 3. 'Classical solutions in Grassmannian o'-models', Lett. Math. Phys. 5 (1981) 553-561. W.-Y. DING

1. 'Lusternik-Schnirelmann theory for harmonic maps', Acta Math. Sinica 2 (1986) 105-122. 2. 'Symmetric harmonic maps between spheres', preprint, Acad. Sinica, 1987. C. T. DODSON, L. VANHECKE and M. E. VAZQuEZ-ABAL

1. 'Harmonic geodesic symmetries', C.R. Math. Rep. Acad. Sci. Canada 9 (1987) 231-235. S. K_ DONALDSON

1. 'Instantons and geometric invariant theory', Comm. Math. Phys. 93 (1984) 453-460. 2. 'The Yang-Mills equations on euclidean space', Perspectives in Math. (Oberwolfach 1944-1984) (W. Jager, J. Moser, R. Remmert, eds., Birkhaliser, 1984), pp. 93-109.

3. 'Twisted harmonic maps and the self-duality equations', Proc. London Math. Soc. 55 (1987) 127-131. H. DONNELLY

1. 'Bounded harmonic functions and positive Ricci curvature', Math. Z. 191 (1986) 559-565. S. DONNINI, G. GIGANTE and L. VANHECKE

1. 'Harmonic reflections with respect to submanifolds', Illinois J. Math., to appear. J. DOUGLAS

1. 'Solutions to the problem of Plateau', Trans. Amer. Math. Soc. 33 (1931) 263-321. 2. 'Some new results in the problem of Plateau', J. Math. Phys. 15 (1936) 55-64. 3. 'Minimal surfaces of higher topological structure', Ann. of Math. 40 (1939) 205-298. G. DZIUK

1. 'Uber die Stetigkeit teilweise freier Minimalflachen', Manuscripta Math. 36 (1981) 241-251. 2. 'C2-regularity for partially free minimal surfaces', Math. Z. 189 (1985) 71-79C. J. EARLE and J EELLS

1. 'Deformations of Riemann surfaces', Lecture Notes in Math. 103 (Springer, Berlin,

1969),

pp. 122-149. 2. 'A fibre bundle description of Teichmuller theory', J. D f Geom. 3 (1969) 19-43. P. EBERLEIN

1. 'When is a geodesic flow of Anosov type? IF, J. Difj: Geom. 8 (1973) 565-577. A. L. EDMONDS

1. 'Deformations of maps to branched coverings in dimension two', Ann. of Math. 110 (1979) 113-125.

191

ANOTHER REPORT ON HARMONIC MAPS

507

J. EELLS

1. 'Minimal graphs', Manuscripta Math. 28 (1979) 101-108. 2. 'On equivanant harmonic maps', Proc. 1981 Shanghai-Hefei Sympos. Du Geom. Diff Eq. (Sci. Press, Beijing, 1984), pp. 55-73. 3. 'Regularity of certain harmonic maps', Global Riemannian Geometry, Durham (1982) (E. Horwood, 1984), pp. 137-147.

4. 'Gauss maps of surfaces', Perspectives in Math. (Oberwolfach 1944-1984) (W. Jager, J. Moser, R. Remmert, eds., Birkhauser, 1984), pp. 111-129. 5. 'Minimal branched immersions into three-manifolds', Proc. Univ. Maryland (1983-1984), Lecture Notes in Math. 1167 (Springer, Berlin, 1985), pp. 81-94. J. EELLS and K. D. ELWORTHY

1. 'On Fredholm manifolds', Actes Congr. Internat. Math. Nice 1970, Vol. 2 (1971), pp. 215-219. J EELLS and L. LEMALRE

1. 'A report on harmonic maps', Bull. London Math. Soc. 10 (1978) 1-68. 2. 'On the construction of harmonic and holomorphic maps between surfaces', Math. Ann. 252 (1980) 27-52. 3. 'Deformations of metrics and associated harmonic maps', Patodi Memorial Vol. Geometry and Analysis (Tata Inst., 1981), pp. 33-45. 4. Selected topics in harmonic maps, C.B.M.S. Regional Conf. Series 50 (Amer. Math. Soc., Providence, R.I., 1983). 5. 'Examples of harmonic maps from disks to hemispheres', Math. Z. 185 (1984) 517-519. J. EELLS and J. C. POLICING

1. 'Removable singularities of harmonic maps', Indiana Univ. Math. J. 33 (1984) 859-871. J. EELLS and A. RATro

1. 'Harmonic maps between spheres and ellipsoids', preprint, I.H.E.S., 1988. J. EELLS and S. SALAMON

1. 'Constructions twistorielles des applications harmoniques', C.R Acad. Sci. Paris 1 296 (1983) 685-687. 2. 'Twistorial constructions of harmonic maps of surfaces into four-manifolds', Ann. Scuola Norm. Sup. Pisa (4) 12 (1985) 589-640. 1. EELLS and J. H. SAMPSON

1. 'Harmonic mappings of Riemannian manifolds', Amer. J. Math. 86 (1964) 109-160. J. EELLs and J. C. WooD

1. 'Restrictions on harmonic maps of surfaces', Topology 15 (1976) 263-266. 2. 'Maps of minimum energy', J. London Math. Soc. (2) 23 (1981) 303-310.

3. 'The existence and construction of certain harmonic maps', Sympos. Math. Rome 26 (1982) 123-138.

4. 'Harmonic maps from surfaces to complex projective spaces', Adv. in Math. 49 (1983) 217-263. N. EnRu

1. 'Totally real submanifolds in a 6-sphere', Proc. Amer. Math. Soc. 83 (1981) 759-763. 2. 'The index of minimal immersions of S2 into SZ"', Math. Z. 184 (1983) 127-132. 3. 'Calabi lifting and surface geometry in S°', Tokyo Math. J. 9 (1986) 297-324. 4. 'Some minimal immersions of spheres into a unit sphere', preprint. 5. 'Isotropic harmonic maps of Riemann surfaces into the de Sitter space time', preprint. H. 1. ELIASSON

1. 'Geometry of manifolds of maps', J. Duff Geom. I (1967) 169-194. K. D. ELWORTHY and W. S. KENDALL

1. 'Factorization of harmonic maps and Brownian motions', Research Notes in Math. 150 (Pitman, 1986), pp. 75-83. S. ERDEM

1. `Harmonic maps from surfaces into pseudo-Riemannian spheres and hyperbolic spaces', Math. Proc. Camb. Phil. Soc. 94 (1983) 483-494. S. ERDEM and J. F. GLAZEBROOK

1. 'Harmonic maps of Riemann surfaces to indefinite complex hyperbolic and projective spaces', Proc. London Math. Soc. 41 (1983) 547-562. S. ERDEM and J. C. WOOD

1. 'On the construction of harmonic maps into a Grassmannian', J. London Math. Soc. (2) 28 (1983) 161-174. J. L. EIUCKSEN

1. 'Equilibrium theory of liquid crystals', Adv. Liquid Cryst. 2 (Academic Press, 1976), pp. 233-298.

192

508

J. FELLS AND L. LEMAIRE

J. H. EsCHENBURG, 1. V. GUADALUPE and R TRIBUZY

1. 'The fundamental equations of minimal surfaces in CP2', Math. Ann 270 (1985) 571-598 J.ESCHENBURG and R. TRIBUZY

1. `Branch points of conformal mappings of surfaces', Math. Ann. 279 (1988) 621-633. A. FATHI, F. LAUDENBACH and V. POENARU (Eds.)

1. Travaux de Thurston sur les surfaces, Asterisque 66-67 (Soc. Math. France, 1979). H. FEDERER

1. 'The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing

flat chains modulo two with arbitrary codimension', Bull. Amer. Math. Soc. 76 (1970) 767-771. C. FEFFERMAN

1. 'The Begman kernel and biholomorphic mappings of pseudoconvex domains', Invent. Math. 26 (1974) 1-65. M. J. FERREIRA

1. 'Morse indices for certain harmonic maps of surfaces', Bull Soc. Math. Belg. B 36 (1984) 131-153. 2. 'A twistorial characterization of conformal branched immersions with parallel mean curvature', Bull. Soc. Math. BeIg. B 39 (1987) 47-81. 3. 'Aplicapoes ramificadas conformer de superffcies de Riemann e problemas variacionais', thesis, University of Lisbon, 1985. D. FERUS and H. KARCHER

1. 'Non-rotational minimal spheres and minimizing cones', Comment. Math Helv. 60 (1985) 247-269. D. FERUS, H. KARCHER and H. F. MUNZNER

1. 'Cliffordalgebren and neue isoparametrische Hyperflachen', Math. Z. 177 (1981) 479-502. R. FINN

1. 'Remarks relevant to minimal surfaces, and surfaces of prescribed mean curvature', J. Analyse Math. 14 (1965) 139-160. A. FISCHER and A. J. TROMBA

I. 'A new proof that Teichmuller space is a cell', Trans. Amer Math. Soc. 303 (1987) 257-262. D. FISCHER-COLBRIE

1. 'On complete minimal surfaces with finite Morse index in three manifolds', Invent. Math. 82 (1985) 121-132. D. FISCHER-COLBRIE and R. SCHOEN

1. 'The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature', Comm. Pure Appl. Math. 33 (1980) 199-211. H. FLANDERS

1. 'Remark on mean curvature', J. London Math. Soc. 41 (1966) 364-366. T. FRANKEL

1. 'On the fundamental group of a compact minimal submanifold', Ann. of Math. 83 (1966) 68-73. D. S. FREED

1. 'The geometry of loop groups', thesis, Mass. Inst. Tech., 1985. 2. 'Flag manifolds and infinite dimensional Kahler geometry', Math. Sci. Res. Inst. Publ., to appear. M. FREEDMAN, J. HAss and P. ScoTr

1. 'Least area incompressible surfaces in 3-manifolds', Invent. Math. 71 (1983) 609-642. T. FRIEDRICH

1. 'On surfaces in four-spaces', Ann. Global Anal. Geom. 2 (1984) 257-287. B. FUGLEDE

1. 'Harmonic morphisms between Riemannian manifolds', Ann. Inst. Fourier (Grenoble) 28 (1978) 107-144

2. 'A criterion of non-vanishing differential of a smooth map', Bull. London Math. Soc. 14 (1982) 98-102. H. FUJIMOTO

1. 'On the number of exceptional values of the Gauss map of minimal surfaces', preprint, Kanazawa University, 1987. A. FUTAKI

1. 'Non-existence of minimizing harmonic maps from 2-spheres', Proc. Japan Acad. 56 (1980) 291-293. 2. 'On the uniqueness of the Dirichlet problem for harmonic maps', J. Fac. Sci. Univ. Tokyo Math. 27 (1980) 181-192. W. D. GARBER, S. N. M. RUUSENAARS, E. SEILER and D. BURNS

1. 'On finite action solutions of the nonlinear a-model', Ann. of Phys. 119 (1979) 305-325.

193

ANOTHER REPORT ON HARMONIC MAPS

509

P. GAUDUCHON

1. 'Pseudo-immersions superminimales d'une surface de Riemann dans une variete riemannienne de dimension 4', Bull. Soc. Math. France 114 (1986) 447-508. P. GAUDUCHON and H. B. LAWSON

1. 'Topologically nonsingular minimal cones', Indiana Univ. Math. J. 34 (1985) 915-927. M GIAQUINTA and E. GIUSTI

1. 'On the regularity of the minima of variational integrals', Acta Math. 148 (1982) 31-46. 2. 'The singular set of the minima of certain quadratic functionals', Ann. Scuola Norm. Sup. Pisa (4) 11 (1984) 45-55. M. GIAQUINTA and S. HILDEBRANDT

1. 'A priori estimates for harmonic mappings', J. Reine Angew. Math 336 (1982) 124-164. M. GIAQUINTA and J. SOUCEK

1. 'Harmonic maps into a hemisphere', Ann. Scuola Norm. Sup. Pisa (4) 12 (1985) 81-90. G. GIGANTE

1. 'A note on harmonic morphisms', preprint, University of Camerino, 1983. V GLASER and R. STORA

1. 'Regular solutions of the CP' models and further generalizations', preprint, CERN, 1980. J. F. GLAZEBROOK

1. 'The construction of a class of harmonic maps to quaternionic projective space', J. London Math. Soc. 30 (1984) 151-159. 2. 'Harmonic maps of Riemann surfaces to indefinite complex Grassmannians and the classical domains', Proc. London Math. Soc. 48 (1984) 108-120. W. B. GORDON

1. 'Convex functions and harmonic maps', Proc. Amer. Math. Soc. 33 (1972) 433-437. H GRAUERT

1. 'On Levi's problem and the imbedding of real-analytic manifolds', Ann. of Math. 68 (1958) 460-472. A. GRAY and J. WOLF

1. 'Homogeneous spaces defined by Lie group automorphisms', J. Dif. Geom. 2 (1968) 77-159. M. A. GRAYSON

1. 'The heat equation shrinks embedded plane curves to round points', J. Dijj: Geom. 26 (1987) 285-314.

2. 'Shortening embedded curves', preprint, University of California, San Diego, 1987. R. E. GREENE and H. H. Wu

1. 'Integrals of subharmonic functions on manifolds of nonnegative curvature', Invent. Math 27 (1974) 265-298.

2. 'Analysis on noncompact Kahler manifolds', Proc Sympos. Pure Math. 30 (1977) 69-100. 3. Function theory on manifolds which possess a pole, Lecture Notes in Math 699 (Springer, Berlin, 1979).

M. GRoMov

1. 'Pseudo holomorphic curves in symplectic manifolds', Invent. Math. 82 (1985) 307-347. A. GROTHENDIECK

1. 'Sur la classification des fibres holomorphes sur la sphere de Riemann', Amer. J. Math. 79 (1957) 121-138. M. GAITER, S. HILDEBRANDT and J. C. C. NITSCHE

1. 'On the boundary behavior of minimal surfaces with a free boundary which are not minima of the area', Manuscripta Math. 35 (1981) 387-410. M. GAITER and J. JOST

1. 'On embedded minimal discs in convex bodies', Ann. Inst. H. Poincard, Anal. Non Lin. 3 (1986) 345-390.

C.-H. Gu 1. 'On the Cauchy problem for harmonic maps defined on two-dimensional Minkowski space', Comm. Pure Appl. Math 33 (1980) 727-737. 2. 'On the initial-boundary value problem for harmonic maps from the 2-dimensional Minkowski space', Manuscripta Math. 33 (1980) 51-58. 3. Some problems on non-linear hyperbolic systems', Proc. 1980 Beijing Sympos. Dif. Geom. Duff Eq. (Gordon and Breach, 1983), Vol. 1, pp. 467-479. 4. 'On the harmonic maps from to S','', J. Reine Angew. Math. 346 (1984) 101-109.

194

510

J. EELLS AND L. LEMAIRE

M. GUEST

1. `Geometry of maps between generalized 2. `The energy of harmonic maps obtained flag manifolds', J. Diff. Geom. 25 (1987) 223-247. 1987.

by the twistor construction', preprint, University of Rochester,

3. `Orbits and harmonic maps', Lefschetz Centenary, Mexico (1984), Contemp. Math. 58 III (1987), pp. 161-171. R. D. GULLIVER

1. `Regularity of minimizing surfaces prescribed mean curvature', Ann. of Math. 97 (1973) 275-305. 2. 'Removability of singular points onofsurfaces of bounded mean curvature', J Dii f. Geom. 11 (1976) 345-350

3. 'Index and total curvature of complete minimal surfaces', Proc. Sympos Pure Math. 44 (1986) 207-211. R. GULLIVER and J. JOST

1. 'Harmonic maps which solve a free boundary problem', prepnnt, University of Minnesota, 1986 R. GULLIVER and H. B. LAWSON

1. `The structure of stable minimal hypersurfaces near a singularity', Proc. Sympos. Pure Math. 44 (1986) 213-237.

R. GULLIVER and B. WHITE

1. 'The rate of convergence of a harmonic map at a singular point', Math. Ann., to appear. R. S. HAMILTON

1. Harmonic maps of manifolds with boundary, Lecture Notes in Math. 471 (Springer, Berlin, 1975). 2. 'Three-manifolds with positive Ricci curvature', J. Diff. Geom. 17 (1982) 255-306. 3. 'Four-manifolds with positive curvature tensor', J. Diff. Geom. 24 (1986) 153-179 R. HARDT and D. KINDERLEHRER

1. `Mathematical questions of liquid crystals', preprint. R. HARDT, D. KINDERLEHRER and F. H. LIN

1. `Energy bounds for minimizing maps', preprint. R. HARDT and F. H. LIN

1. 'Mappings minimizing the L" norm of the gradient', preprint, Austral. Nat. Univ., 1986. R. HARDT and L. SIMON

1. `Boundary regularity and embedded solutions for the oriented Plateau problem', Ann. of Math. 110 (1979) 439-486. P. HARTMAN

1. 'On homotopic harmonic maps', Canad. J. Math. 19 (1967) 673-687. H. HEFrER

1. `Dehnungsuntersuchungen an Spharenabbildungen', Invent. Math. 66 (1982) 1-10. E. HEINZ

1. `Uber Fla"chen mit eineindeutigen Projektion auf eine Ebene, deren Kriimmung durch Ungleichungen

eingeschrankt rind', Math. Ann. 129 (1955) 451-454. E. HEINZ and S. HILDEBRANDT 1.

Some remarks on minimal surfaces in Riemannian manifolds', Comm. Pure App!. Math. 23 (1970) 371-377.

F. HaLEIN

1. 'Regularity and uniqueness of harmonic maps into an ellipsoid', preprint, Ecole Polytechnique, Paris.

2.

' Homeomorphismes quasi-conformes entre surfaces Riemanniennes', preprint, Ecole Polytechnique, Paris.

S. HELGASON

1. Differential geometry, Lie groups, and symmetric spaces (Academic Press, 1978). H. HESS, R SCHRADER and D. A. UHLENBROCK

1. 'Kato's inequality and the spectral distribution of Laplacians on compact Riemannian manifolds',

J. Dif. Geom. 15 (1980) 27-37. S HILDEBRANDT

1. 'Liouville theorems for harmonic mappings, and an approach to Bernstein theorems', Ann, of Math. Studies 102 (1982) 107-131. 2. `Nonlinear elliptic systems and harmonic mappings', Proc. Beijing Sympos. Diff. Geom. Diff. Eq. (1980) (Gordon and Breach, 1983), pp- 481-615. 3. 'Quasilinear elliptic systems in diagonal form', Systems of nonlinear partial differential equations (J. M.

Ball, ed., Reidel, Dordrecht, 1983), pp. 173-217.

195

ANOTHER REPORT ON HARMONIC MAPS

511

4. 'Harmonic mappings of Riemannian manifolds', Harmonic Maps and Minimal Immersions (Montecatini 1984), Lecture Notes in Math. 1161 (Springer, Berlin, 1985), pp. 1-117. S. HILDEBRANDT, J. JOST and K-O. WIDMAN

1. 'Harmonic mappings and minimal submanifolds', Invent. Math. 62 (1980) 269-298. S. HILDEBRANDT, H. KAUL and K-O. WIDMAN

1. 'An existence theorem for harmonic mappings of Riemannian manifolds', Acta Math. 138 (1977) 1-16. M. HIRSCH

1. 'Immersions of manifolds', Trans. Amer. Math. Soc. 93 (1959) 242-276. N. HITCHIN

1. 'Kahlerian twistor spaces', Proc. London Math. Soc. (3) 43 (1981) 133-150. 2. 'Harmonic maps from a 2-torus to the 3-sphere', preprint, Oxford, 1987. 3. 'The self-duality equations on a Riemann surface', Proc. London Math. Soc. (3) 55 (1987), 59-126. D A. HOFFMAN and W. H. MEEKS III

1. 'Complete embedded minimal surfaces of finite total curvature', Bull. Amer. Math. Soc. 12 (1985) 134-136.

2. 'A complete embedded minimal surface in R3 with genus one and three ends', J. Diff. Geom. 21 (1985) 109-127. D. A. HOFFMAN and R. OSSERMAN

1. The geometry of the generalized Gauss map, Mem. Amer. Math. Soc. 236 (1980). D. A. HOFFMAN, R. OSSERMAN and R. SCHOEN

1. 'On the Gauss map of complete surfaces of constant mean curvature in 683 and R4', Comment. Math. Hely. 57 (1982) 519-531. H. Hope

1. 'Uber Flachen mit einer Relation zwischen den Hauptkriimmungen', Math. Nachr. 4 (1950-51) 232-249. A. HOWARD and B. SMYTH

1. 'Kahler surfaces of nonnegative curvature', J. Df. Geom. 5 (1971) 491-502. A. HOWARD, B. SMYTH and H. H Wu

1. 'On compact Kahler manifolds of nonnegative bisectional curvature I', Acta Math. 147 (1981) 51-56. R. HOWARD

1. 'The non-existence of stable submanifolds, varifolds, and harmonic maps in sufficiently pinched simply connected Riemannian manifolds', Michigan Math J. 32 (1985) 321-334. R. HOWARD and W. WEI

1. 'Nonexistence of stable harmonic maps to and from certain homogeneous spaces and submanifolds of euclidean space ', Trans. Amer. Math. Soc. 294 (1986) 319-331. W. T. HSIANG and W. Y. HSIANG

1. 'Examples of codimension-one closed minimal submanifolds in some symmetric spaces, I', J. Dii f. Geom. 15 (1980) 543-551.

2. 'On the existence of codimension-one minimal spheres in compact symmetric spaces of rank 2, IF, J. Dii f. Geom. 17 (1982) 583-594.

3. 'An infinite family of minimal imbeddings of Sea-1 into S"(1) x S"(1), n = 2,3', preprint. 4. 'On the construction of codimension two minimal immersions of exotic spheres into Euclidiean spheres, IF, Math Z. 195 (1987) 301-313. W. T. HSIANG, W. Y. HSIANG and I. STERLING

1. 'On the construction of codimension two minimal immersions of exotic spheres into Euclidean spheres', Invent. Math. 82 (1985) 447-460. W. T. HSIANG, W. Y. HSIANG and P. TOMTER

1. 'On the construction of infinitely, many mutually noncongruent, examples of minimal embeddings of Ss"-1 into CP", n _> 2', Bull. Amer. Math. Soc. 8 (1983) 463-465. W. Y. HSIANG

1. 'On the compact homogeneous minimal submanifolds', Proc. Nat. Acad. Sci. USA 56 (1966) 5-6. 2. ' Remarks'on closed minimal submanifolds in the standard Riemannian m-sphere', J. Dii f. Geom. 1 (1967) 257-267.

3. 'On generalizations of theorems of A. D Alexandrov and C. Delaunay on hypersurfaces of constant mean curvature', Duke Math. J. 49 (1982) 485-496. 4. 'Generalized rotational hypersurfaces of constant mean curvature in the euclidean spheres, I', J. Diff. Geom. 17 (1982) 337-356.

196

J. EELLS AND L. LEMAIRE

512

5. `Minimal cones and the spherical Bernstein problem, I', Ann. of Math. 118 (1983) 61-73. 6. 'Minimal cones and the spherical Bernstein problem, II', Invent. Math. 74 (1983) 351-369. W. Y. HSIANG and H. B. LAWSON

1. 'Minimal submanifolds of low cohomogeneity', J Dil Geom 5 (1971) 1-38. W. Y. HSIANG and 1. STERLING

1. 'On the construction of non-equatorial minimal hypersurfaces in S"(1) with stable cones in 68 "'r'. Proc. Nat. Aced Sci. USA 81 (1984) 8035-8036. W. Y. HSIANG, Z. H. TENG and W.-C. Yu 1. 'Examples of constant mean curvature immersions of 3-sphere into euclidean 4-space', Proc. Nat. Acad Sci. USA 79 (1982) 3931-3932.

2. 'New examples of constant mean curvature immersions of (2k - 1) spheres into Euclidean 2k-space', Ann. of Math. 117 (1983) 609-625. W. Y. HSIANG and W. Yu

1. 'A generalization of a theorem of Delaunay'. J. Dif. Geom. 16 (1981) 161-177. Hu HESHENo

1. 'A nonexistence theorem for harmonic maps with slowly divergent energy', Chin. Ann. Math 5B (1984) 737-740. T. ISHIHARA

1. 'The index of a holomorphic mapping and the index theorem', Proc Amer. Math. Soc. 66 (1977) 169-174.

2. 'A mapping of Riemannian manifolds which preserves harmonic functions', J. Math. Kyoto Univ 19 (1979) 215-229.

3. 'Harmonic sections of tangent bundles', J. Math. Tokushima Univ. 13 (1979) 23-27. 4. 'The harmonic Gauss maps in a generalized sense', J. London Math. Soc. 26 (1982) 104-112. W. JAGER

1. 'Behavior of minimal surfaces with free boundaries'. Comm. Pure Appl. Math. 23 (1970) 803-818. W. JAGER and H. KAUL

1. 'Uniqueness and stability of harmonic maps, and their Jacobi fields', Manuscripta Math. 28 (1979) 269-291. 2.' Uniqueness of harmonic mappings and of solutions of elliptic equations on Riemannian manifolds', Math. Ann. 240 (1979) 231-250. 3. 'Rotationally symmetric harmonic maps from a ball into a sphere and the regularity problem for weak solutions of elliptic systems', J. Reine Angew Math. 343 (1983) 146-161. H. JENKINS and J. SERRIN

1. 'The Dirichlet problem for the minimal surface equation in higher dimensions', J. Reine Angew. Math. 229 (1968) 170-187. G. JENSEN and M. RIGOLI

1. 'Harmonically immersal surfaces of B", preprint, I.C.T.P. Trieste, 1987. 2. 'Harmonic Gauss maps', Preprint, I.C.T.P. Trieste, 1987. M. JI

1. 'Minimal surfaces in Riemannian manifolds', thesis, Acad. Sinica, 1987. P. W. JONES

1. 'A complete bounded complex submanifold of C1', Proc. Amer. Math. Soc. 76 (1979) 305-306. L. JORGE and F XAVIER

1. 'A complete minimal surface in R1 between two parallel planes', Ann. of Math. 112 (1980) 204-206. 2. 'An inequality between the exterior diameter and the mean curvature of bounded immersions', Math Z. 178 (1981) 77-82. J. JOST

1. 'Univalency of harmonic mappings between surfaces', J. Reine Angew. Math. 324 (1981) 141-153. 2. 'Eine geometrische Bemerkung zur Satzen Ober harmonische Abbildungen, die ein Dirichletproblem losen', Manuscripla Math. 32 (1980) 51-57. 3. ' Ein Existenzbeweis fur harmonische Abbildungen, die ein Dirichletproblem losen, mittels der

Methode des Warmeflusses', Manuscripta Math. 34 (1981) 17-25. 4. 'A maximum principle for harmonic mappings which solve a Dirichlet problem', Manuscripta Math. 38 (1982) 129-130. 5. 'Existence proofs for harmonic mappings with the help of a maximum principle', Math. Z. 184 (1983) 489-496.

6. 'The Dirichlet problem for harmonic maps from a surface with boundary onto a 2-sphere with nonconstant boundary values', J. Diff. Geom. 19 (1984) 393-401. 7. Harmonic mappings between surfaces, Lecture Notes in Math. 1062 (Springer, Berlin, 1984).

197

513

ANOTHER REPORT ON HARMONIC MAPS

8. Harmonic mappings between Riemannian manifolds, Proc. Centre Math. Analysis (Aust. Nat. Univ. Press, Canberra, 1983). 9. 'Conformal mappings and the Plateau-Douglas problem in Riemannian manifolds', J. Reine Angew. Math. 359 (1985) 37-54. 10. 'On the existence of harmonic maps from a surface into the real projective space', Compositio Math. 59 (1986) 15-19.

11. 'A note on harmonic maps between surfaces', Ann. Inst. H. Poincare Anal. Non Lin. 2 (1985) 397-405.

12. 'Lectures on harmonic maps', Harmonic Mappings and Minimal Immersions (Montecatini 1984), Lecture Notes in Math. 1161 (Springer, Berlin, 1985), pp. 118-192. 13. 'Existence results for embedded minimal surfaces of controlled topological type. I, 11, III', Ann. Scuola Norm. Sup Pisa 13 (1986) 15-50; 401-426. 14. 'On the regularity of minimal surfaces with free boundaries in Riemannian manifolds', Manuscripta Math. 56 (1986) 279-291. J. JosT and H. KARCHER

1. 'Geometrische Methoden zur Gewinnung von a priori-Schranken fur harmonische Abbildungen', Manuscripta Math. 40 (1982) 27-77. 2. 'Almost linear functions and a-priori estimates for harmonic maps', Global Riemannian Geom., Durham (1982) (E. Horwood, 1984), pp. 148-155. J. JOST and M MEIER

1. 'Boundary regularity for minima of certain quadratic functionals', Math. Ann. 262 (1983) 549-561. J. JosT and R. SCHOEN

1. 'On the existence of harmonic diffeomorphisms between surfaces', Invent. Math. 66 (1982) 353-359. J. JosT and S. T. YAU 1. 'Harmonic mappings and Kahler manifolds', Math. Ann. 262 (1983) 145-166. 2. 'A strong rigidity theorem for a certain class of compact complex analytic surfaces', Math. Ann. 271 (1985) 143-152. 3. 'The strong rigidity of locally symmetric complex manifolds of rank one and finite volume', Math. Ann. 275 (1986) 291-304. 4. 'On the rigidity of certain discrete groups and algebraic varieties', Math. Ann. 278 (1987) 481-496. M. KALKA

1. 'Deformation of submanifolds of strongly negatively curved manifolds', Math. Ann. 251 (1980) 243-248. N. KAPOULEAs

1. 'Constant mean curvature surfaces in Euclidean three-space', Bull. Amer. Math. Soc. 17 (1987) 318-320. H. KARCHER, U. PINKALL and I. STERLING

1. 'New minimal surfaces in S3', preprint, Max-Planck Institute, 1986. H. KARCHER and J. C. WooD

1. 'Non-existence results and growth properties for harmonic maps and forms', J. Reine Angew. Math. 353 (1984) 165-180. L. KARP

1. 'Subharmonic functions on real and complex manifolds', Math. Z. 179 (1982) 535-554. 2. 'Subharmonic functions, harmonic mappings, and isometric immersions', Ann. of Math. Studies 102 (1982) 133-142. 3. 'Differential inequalities on complete Riemannian manifolds and applications', Math. Ann. 272 (1985) 449-459. A. KASUE

1. 'Harmonic functions with growth conditions on a manifold with asymptotically nonnegative curvature I, IF, preprints, Osaka University. W. S. KENDALL

1. 'Brownian motion and a generalised little Picard's theorem', Trans. Amer. Math. Soc. 275 (1983) 751-760.

2. 'Stochastic differential geometry: An introduction', Acta Appl. Math. 9 (1987) 29-60.

3. 'Martingales on manifolds and harmonic maps', Geom. of Random Motion, Amer. Math. Soc. Contemp. Math., to appear. K. KENMOTSU

1. 'Weierstrass formula for surfaces of prescribed mean curvature', Math. Ann. 245 (1979) 89-99. F. C. KIRWAN

1. Cohomology of quotients in symplectic and algebraic geometry, Math. Notes 31 (Princeton University Press, 1984). 17

BLM 20

198

J. FELLS AND L. LEMAIRE

514 S. KORAYASxt and K. NoMizu

1. Foundations of differential geometry (Interscience, 1963, 1969). N. Koiso

1. 'Variation of harmonic mappings caused by a deformation of Riemannian metrics', Hokkaido Math. J. 8 (1979) 199-213. N. KOREVAAR, R. KUSNER and B. SOLOMON

1. 'The structure of complete embedded surfaces with constant mean curvature', preprint, University of California, San Diego. J. L. KOSZUL and B. MALGRANGE

1. 'Sur certaines structures fibrees complexes', Arch. Math. (Basel) 9 (1958) 102-109. 0. LADYZENSKAYA and N. URAL'CEVA

1. Linear and quasilinear elliptic equations (Academic Press, 1968). H. B. LAWSON

1. 'Local rigidity theorems for minimal hypersurfaces', Ann. of Math. 89 (1969) 187-197. 2. 'The global behavior of minimal surfaces in S"', Ann. of Math. 92 (1970) 224-237. 3. Lectures on minimal submanifolds, Vol. 1 (IMPA, Rio de Janeiro, 1970; second edition, Publish or Perish, 1980).

4. 'Complete minimal surfaces in S3', Ann. of Math. 92 (1970) 335-374 5. 'The unknottedness of minimal embeddings', Invent. Math. 11 (1970) 183-187. 6. 'Surfaces minimales et is construction de Calabi-Penrose', Sim. Bourbaki 624 (1983/4), Asterisque 121-122 (Soc. Math. France, 1985), pp. 197-211. H. B. LAWSON and R. OSSERMAN

1. 'Non-existence, non-uniqueness and irregularity of solutions to the minimal surface system', Acta Math. 139 (1977) 1-17. H. B. LAWSON and J. SIMONS

1. 'On stable currents and their application to global problems in real and complex geometry', Ann. of Math. 98 (1973) 427-450. C. LEBRUN

1. 'Twistor CR manifolds and three-dimensional conformal geometry', Trans. Amer. Math Soc. 284 (1984) 601-616. M.-L. LEITE

1. 'Harmonic mappings of surfaces with respect to degenerate metrics', Amer. J. Math. 110 (1988) 399-412. J.LELONG-FERRAND

1. 'Construction de modules de continuite dans le cas limite de Soboleff et applications a la g6ometrie difffirentielle', Arch. Rat. Mech. Anal. 52 (1973) 297-311. L. LEMAIRE

1. 'Applications harmoniques de varietes produits', Comment. Math. Hely. 52 (1977) 11-24. 2. 'Applications harmoniques de surfaces riemanniennes', J. Diff. Geom. 13 (1978) 51-78.

3. 'Harmonic nonholomorphic maps from a surface to a sphere', Proc. Amer. Math. Soc. 71 (1978) 299-304. 4. 'Existence des applications harmoniques et courbure des varietes', Sem. Bourbaki 553 (1980), Lecture Notes in Math. 842 (Springer, Berlin, 1981), pp. 174-195. 5. 'Boundary value problems for harmonic and minimal maps of surfaces into manifolds', Ann. Scuola Norm. Sup. Pisa (4) 9 (1982) 91-103. 6. 'Harmonic maps of finite energy from a complete surface to a compact manifold', Sympos. Math. 26 (Bologna, 1982), pp. 23-26. P. F. LEUNG

1. 'On the stability of harmonic maps', Harmonic Maps Proc. Tulane, Lecture Notes in Math. 949 (Springer, Berlin, 1982), pp. 122-129. 2. 'Minimal submanifolds in a sphere', Math. Z. 183 (1983) 75-86. 3. 'A note on stable harmonic maps', J. London Math. Soc. 29 (1984) 380-384. H. LEwY

1. 'On minimal surfaces with partially free boundaries', Comm. Pure Appl. Math. 4 (1951) 1-13. 2. 'On the non-vanishing of the jacobian of a homeomorphism by harmonic gradients', Ann. of Math. 88 (1968) 518-529. P. Li

1. 'Uniqueness of L' solutions for the Laplace equation and the heat equation on Riemannian manifolds', J. Diff. Geom. 20 (1984) 447-457.

199

515

ANOTHER REPORT ON HARMONIC MAPS P. Li and R. SCHOEN

1. 'LP and mean value properties of subharmonic functions on Riemannian manifolds', Acta Math. 153 (1984) 279-301. P. Li and L.-F. TAM

1. 'Positive harmonic functions on complete manifolds with non-negative curvature outside a compact set', Ann. of Math. 125 (1987) 171-207. A. LiCHNEROWICZ

1. 'Applications harmoniques et varietes kahleriennes', Sympos. Math. 3 (Bologna, 1970), pp. 341-402. S. LUCKHAUS

1. 'Partial Holder continuity for energy minimizing p-harmonic maps between Riemannian manifolds', preprint. G MARGULIS

1. 'Applications of ergodic theory to the investigation of manifolds of negative curvature', Funct. Anal. Appl. 3 (1969) 335-336. K. MASHIMo

1. 'Minimal immersions of 3-dimensional sphere into spheres', Osaka J. Math. 21 (1984) 721-732. A. MAYER

1. 'Families of K-3 surfaces', Nagoya Math. J. 48 (1972) 1-17 P. O. MAZUR

1. 'Proof of uniqueness of the Kerr-Newman black hole solution', J. Phys. A 15 (1982) 3173-3180. W. H. MEEKS

1. 'The conformal structure and geometry of triply periodic minimal surfaces in lv', thesis, University of California, Berkeley, 1975. 2. 'A survey of the geometric results in the classical theory of minimal surfaces', Bol. Soc. Bras. Mat. 12 (1981) 29-86. 3. 'Recent progress on the geometry of surfaces in ll and on the use of computer graphics as a research tool', Proc. Internat. Cong. Math. Berkeley, 1986 (Amer. Math Soc., 1987), pp. 551-560. 4. 'The topology and geometry of embedded surfaces of constant mean curvature', Bull. Amer. Math. Soc. 17 (1987) 315-317; J. Dif. Geom. 27 (1988) 539-552. 5. 'The topological uniqueness of minimal surfaces in three dimensional Euclidean space', Topology 20 (1981) 389-410. W. H. MEEKS, L. SIMON and S. T. YAU

1. 'Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature', Ann. of Math. 116 (1982) 621-659. W. H. MEEKS and S. T. YAU

1. 'Topology of three dimensional manifolds and the embedding problems in minimal surface theory', Ann. of Math. 112 (1980) 441-484. 2. 'The equivariant Dehn's lemma and loop theorem', Comment. Math. Hely. 56 (1981) 225-239. 3. 'The classical Plateau problem and the topology of three dimensional manifolds', Topology 21 (1982) 408-442.

4. 'The existence of embedded minimal surfaces and the problem of uniqueness', Math. Z. 179 (1982) 151-168.

5. 'Group actions on R3', The Smith conjecture (Academic Press, 1984), pp. 167-179. M. MEIER

1. 'Liouville theorems for nonlinear elliptic equations and systems', Manuscripta Math. 29 (1979) 207-228.

2. 'Boundedness and integrability properties of weak solutions of quasilinear elliptic systems', J. Reine Angew. Math. 333 (1982) 191-220 3. 'Removable singularities for weak solutions of quasilinear elliptic systems', J. Reine Angew. Math. 344 (1983) 87-101. 4. 'Removable singularities of harmonic maps and an application to minimal submanifolds', Indiana Univ. Math. J. 35 (1986) 705-726. 5. 'Asymptotic behavior of solutions of some quasilinear elliptic systems in exterior domains', preprint. P. A. MEYER 1.

' Geometrie stochastique sans larmes', Sim. Prob. XV, Lecture Notes in Math 850 (Springer, Berlin, 1981), pp. 44-102.

M. J. MICALLEE

1. 'Stable minimal surfaces in Euclidean space', J. Diff Geom. 19 (1984) 57-84. 2. 'Stable minimal surfaces in flat tori', Contemp. Math. 49 (1986) 73-78. 17-2

200

516

J. EELLS AND L. LEMAIRE

M. MICALLEI' and J. D MooRE

1. 'Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic twoplanes', Ann. of Math. 127 (1988) 199-227. M. MICALLEF and B. WHITE

I. 'On the structure of branch points of minimizing disks', preprint. T. K. MILNOR

1. 'Harmonic maps and classical surface theory in Minkowski 3-space', Trans. Amer. Math. Soc. 280 (1983) 161-185. 2. 'Characterizing harmonic immersions of surfaces with indefinite metric', Proc. Nat Acad. Sci. USA 79 (1982) 2143-2144. MIN-0o

1. 'Maps of minimum energy from compact simply-connected Lie groups', Ann. Global Anal. Geom. 2 (1984) 119-128. N. MOK

1. 'The holomorphic or antiholomorphic character of harmonic maps into irreductible compact quotients of polydiscs', Math. Ann. 272 (1985) 197-216. 2. 'The uniformization theorem for compact Kahler manifolds of nonnegative holomorphic bisectional curvature', J. Dif. .. Geom. 27 (1988) 179-214. N. MOK and J.-Q. ZHONG

1. 'Curvature characterization of compact Hermitian symmetric spaces', J. Dif. . Geom. 23 (1986) 15-67. J. D. MooRE 1. 'Isometric immersions of space forms in space forms', Pacific J. Math. 40 (1972) 157-166.

2. 'On stability of minimal spheres and a two-dimensional version of Synge's theorem', Arch. Math (Basel) 44 (1985) 278-281. 3. 'Compact Riemannian manifolds with positive curvature operators', Bull. Amer. Math. Soc. 14 (1986) 279-282. S. MORI

1. 'Projective manifolds with ample tangent bundles', Ann. of Math. (2) 110 (1979) 593-606. C. B. MORREY

1. 'The problem of Plateau on a Riemannian manifold', Ann. of Math. 49 (1948) 807-851. 2. 'The analytic embedding of abstract real-analytic manifolds', Ann. of Math. 68 (1958) 159-201. 3. Multiple integrals in the calculus of variations, Grundlehren 130 (Springer, Berlin, 1966). M. MORSE and C. B. TOMPKINS

1. 'The existence of minimal surfaces of general cntical types', Ann, of Math. 40 (1939) 443-472. 2. 'Minimal surfaces not of minimum type by a new method of approximation', Ann. of Math. 42 (1941) 62-72. 3. 'Unstable minimal surfaces of higher topological structure', Duke Math. J. 8 (1941) 350-375. J. Moses

1. 'On Harnack's theorem for elliptic differential equations', Comm. Pure Appl. Math. 14 (1961) 577-591. G. D. MosTOW

1. Strong rigidity of locally symmetric spaces, Ann. of Math. Studies 78 (Princeton University Press, 1973).

G. D. Moscow and Y.-T. Slu 1. 'A compact Kahler surface of negative curvature not covered by the ball', Ann. of Math. 112 (1980) 321-360. 0. MuaKAROv

1. 'Structures presques hermitiennes sur des espaces twistoriels et leurs types', C.R. Acad. Sci. Paris A 305 (1987) 307-309. H. NAITO

1. 'Asymptotic behavior of solutions to Eells-Sampson equations near stable harmonic maps', preprint, Nagoya University, 1987. 2. 'On the holomorphicity of pluriharmonic maps', preprint, Nagoya University, 1988. S. NISHIKAWA

1. 'On the Neumann problem for the nonlinear parabolic equation of Eells-Sampson and harmonic mappings', Math. Ann. 249 (1980) 177-190. S. NISHIKAWA and K. SHIGA

1. 'On the holomorphic equivalence of bounded domains in complete Kahler manifolds of nonpositive curvature', J. Math. Soc. Japan 35 (1983) 273-278.

201

ANOTHER REPORT ON HARMONIC MAPS

517

J. C. C. NITscim 1. Vorlesungen fiber Minimalflachen, Grundlehren 199 (Springer, Berlin, 1975). 0.NOUHAUD

1. 'Applications harmoniques d'une variete riemannienne dans son fibre tangent', C.R. Acad. Sci. Paris 1284 (1977) 815-818. M. OBATA

1. 'The Gauss map of immersions of Riemannian manifolds in spaces of constant curvature', J. Diff. Geom. 2 (1968) 217-223. N. R. O'BRIAN and J. H. RAWNSLEY

1. 'Twistor spaces', Ann. Global Anal. Geom. 3 (1985) 29-58. Y. OHNITA

1. 'Stability of harmonic maps and standard minimal immersions', Tbhoku Math. J. 38 (1986) 259-267.

2. 'On pluriharmonicity of stable harmonic maps', J. London Math. Soc. (2) 35 (1987) 563-568. Y. OHNITA and S. UDAGAWA

1. 'Stable harmonic maps from Riemann surfaces to compact Hermitian symmetric spaces', Tokyo Math. J 10 (1987) 385-390.

2. 'Stability, complex-analyticity and constancy of pluriharmonic maps from Kaehler manifolds', preprint, Max-Planck Institute, 1988. H. OMORI

1. 'Isometric immersions of Riemannian manifolds', J. Math. Soc. Japan 19 (1967) 205-214. R. OSSERMAN

1. 'Minimal surfaces in the large', Comment. Math. Hely. 35 (1961) 65-76. 2. 'A proof of the regularity everywhere of the classical solution to Plateau's problem', Ann. of Math. 91 (1970) 550-569. 3. 'On Bets' theorem on isolated singularities', Indiana Univ. Math. J. 23 (1973) 337-342. S. K. OTrARSSON

1. 'Closed geodesics on Riemannian manifolds via the heat flow', J. Geom. Phys. 2 (1985) 49-72. R. S. PALMS

1. 'The principle of symmetric criticality', Comm. Math. Phys. 69 (1979) 19-30. B PALMER

1. Stanford Thesis (1987). Y-L. PAN and Y-B. SHEN

1. 'Stability of harmonic maps and minimal immersions', Proc. Amer. Math. Soc. 93 (1985) 111-117. M. PARKER

1. 'Orthogonal multiplications in small dimensions', Bull. London Math. Soc. 15 (1983) 368-372. W. PARRY and M. PoLLICOTT

1. 'An analogue of the prime number theorem for closed orbits of axiom A flows', Ann of Math. 118 (1983) 573-591. V. PETTINATI and A. RArro

1. 'Existence and non-existence results for harmonic maps between spheres', preprint, University of Warwick, 1987. B. PIETTE and W. J. ZAKRZEWSKI

1. 'General solution of the U(N) chiral a models in two dimensions', preprint. 2. 'Properties of classical solutions of the U(N) chiral a models in two dimensions', preprint. J. T. PITTS

1. Existence and regularity of minimal surfaces on Riemannian manifolds, Math. Notes 27 (Princeton University Press, 1981). 2. 'The index of instability of minimal surfaces obtained by variational methods in the large', preprint. J. T. PIrrS and J. 11. RUBINSTEIN

1. 'Existence of minimal surfaces of bounded topological type in three-manifolds', Proc. Centre Math. Anal. Canberra 10 (1986) 163-176. 2. 'Applications of minimax to minimal surfaces and the topology of 3-manifolds', Proc. Centre Math. Anal. Canberra, to appear. 3. 'Equivariant minimax and minimal surfaces in geometric three-manifolds', preprint. A. I. PLUZHNIKOV

1. 'Harmonic mappings of Riemann surfaces and foliated manifolds', Mat. Sb. (N.S.) 113 (1980) no. 7 (90) 339-347, 352 (Russian); English translation, Math. USSR. Sb. 41 (1982) 281-287.

202

518

J. EELLS AND L. LEMAIRE

2. 'Some properties of harmonic mappings in the case of spheres and Lie groups', Sov. Math. Dokl. 27 (1983) 246-248.

3. 'On the minimum of the Dirichlet functional', Dokl. Akad. Nauk 290 (2) (1986) 289-293 (Russian), English translation, Sov. Math. Dokl. 34 (1987) 281-284. 4. 'A topological criterion for the attainability of global minima of an energy function', Nov. Glob Anal Voronezh. Gos. Univ. 177 (1986) 149-155 Y. S. POON 1.

'Minimal surfaces in four dimensional manifolds', M.Sc thesis, Oxford, 1983.

A. N. PRESSLEY

1. 'The energy flow on the loop space of a compact Lie group', J. London Math. Soc. 2 (26) (1982) 557-566.

2. 'Decomposition of the space of loops on a Lie group', Topology 19 (1980) 65-79. A. N. Prsst.EY and G. SEGAL 1. Loop groups, Oxford Math. Monographs (Clarendon, Oxford, 1986). P PRICE

1. 'A monotonicity formula for Yang-Mills fields', Manuscripta Math. 43 (1983) 131-166. T. RAno 1. 'On the problem of least area and the problem of Plateau', Math. Z. 32 (1930) 763-796. J. RAMANATHAN

1. 'Harmonic maps from S2 to G3 4', J. Di . Geom. 19 (1984) 207-219. 2. 'A remark on the energy of harmonic maps between spheres', Rocky Mountain J. Math. 16 (1986) 783-790. A. RATTo

1. 'On harmonic maps between Ss and Sz of prescribed Hopf invariant', Math. Proc. Cambridge Phil. Soc., to appear. 2. 'Harmonic maps from deformed spheres to spheres', Amer. J. Math., to appear. 3. 'Construction d'applications harmoniques de spheres euclidiennes', C.R. Acad. Sci. Paris 1304 (1987) 185-186.

4. 'Harmonic maps of spheres and equivariant theory', thesis, University of Warwick, 1987 5. 'Equivariant harmonic maps between manifolds with metrics of (p, q)-signature', preprint, I H.E.S., 1988.

J. H. RAwNst.EY

1. 'Noether's theorem for harmonic maps', Dii f Geom. Methods in Math. Phys. (Reidel,

1984),

pp. 197-202. 2. 'f-structures, f-twistor spaces and harmonic maps', Sem. Geom. L. Bianchi 111984, Lecture Notes in Math. 1164 (Springer, Ber}in, 1985), pp. 85-159. 3. 'Harmonic 2-spheres', Coll. Theories Quantiques et Geomdtries, Les Treilles 1987, to appear. M. RtGOLI

1. 'The harmonicity of the spherical Gauss map', Bull. London Math. Soc 18 (1986) 609-612 2. 'The conformal Gauss map of submanifolds of the M6bius space', preprint, I.C.T.P. Trieste, 1987. M. RIGOU and R. TRIBUZY

1. 'The Gauss map for Kahlerian submanifolds of R", preprint, I.C.T.P. Trieste, 1987. J. H. RUBINSTEIN

1. 'Embedded minimal surfaces in 3-manifolds with positive scalar curvature', Proc. Amer. Math. Soc. 95 (1985) 458-462. E. A. RUH 1. 'Minimal immersions of 2-spheres in S4', Proc. Amer. Math. Soc. 28 (1971) 219-222. E. A. RUH and J. VILMS

1. 'The tension field of the Gauss map', Trans. Amer. Math Soc. 149 (1970) 569-573. J. SACKS and K. UHLENBECK

1. 'The existence of minimal immersions of 2-spheres', Ann. of Math. 113 (1981) 1-24. 2. 'Minimal immersions of closed Riemann surfaces', Trans. Amer. Math. Soc. 271 (1982) 639-652. S. SALAMON

1. 'Harmonic and holomorphic maps', Sem. Geom. L. Bianchi 11 1984, Lecture Notes in Math. 1164 (Springer, Berlin, 1985), pp 161-224. 2. 'Minimal surfaces and symmetric spaces', Differential Geometry, Research Notes in Math. 131 (Pitman, 1985), pp. 103-114. 3. 'Degrees of minimal surfaces in 4-manifolds', Luminy Conf. Harmonic Maps, 1986. (World Sci. Press,

to appear.)

203 ANOTHER REPORT ON HARMONIC MAPS

519

1. SALAVESSA

1. 'Graphs with parallel mean curvature and a variational problem in conformal geometry', thesis, University of Warwick, 1987. J. H. SAMPSON

1. 'Some properties and applications of harmonic mappings', Ann. Sci. Ecole Norm. Sup. 11 (1978) 211-228.

2. 'On harmonic mappings', Sympos. Math. 26 (Bologna, 1982), pp. 197-210. 3. 'Harmonic maps in Kahler geometry', Harmonic Maps and Minimal Immersions, CIME Conj. 1984, Lecture Notes in Math. 1161 (Springer, Berlin, 1985), pp. 193-205. 4. 'Applications of harmonic maps to Kahler geometry', Contemp. Math 49 (1986) 125-134. A. SANINI

1. 'Applicazioni tra variety riemanniane con energia critica rispetto a deformazioni di metriche', Rend. Math (7) 3 (1983) 53-63. 2. 'Applicazioni armoniche tra i fibrati tangenti di varieta riemanniane', Bol. Unione Mat. Ital. (6) 2 A (1983) 55-64. L. SARIO, M. NAKAI, C. WANG and L. O. CHUNG

1. Classification theory of Riemannian manifolds, Lecture Notes in Math. 605 (Springer, Berlin, 1977). L. SARIO, M SCHIFFER and M. GLASNER

1. 'The span and principal functions in Riemannian spaces', J. Analyse Math. 15 (1965) 115-134. R. SCHOEN

1. 'Analytic aspects of the harmonic map problem', Math. Sci. Res. Inst. Publ. 2 (Springer, Berlin, 1984), pp. 321-358. 2. 'Estimates for stable minimal surfaces in three dimensional manifolds', Sem. Minimal Submanifolds, Ann. of Math. Studies 103 (E. Bombieri, ed., Princeton University Press, 1983), pp. 111-126. R. SCHOEN and L. SIMON

1. 'Regularity of embedded simply connected minimal surfaces', Ann. of Math., to appear. R. SCHOEN and K. UHLENBECK

1. 'A regularity theory for harmonic maps', J. Difj: Geom. 17 (1982) 307-335 and 18 (1983) 329.

2. `Boundary regularity and the Dirichlet problem for harmonic maps', J. Di/f. Geom. 18 (1983) 253-268.

3. 'Regularity of minimizing harmonic maps into the sphere', Invent. Math. 78 (1984) 89-100. R. SCHOEN and S. T. YAU

1. 'Harmonic maps and the topology of stable hypersurfaces and manifolds of non-negative Ricci curvature', Comment. Math. HeIv. 51 (1976) 333-341. 2. 'On univalent harmonic maps between surfaces', Invent. Math. 44 (1978) 265-278. 3. 'Existence of incompressible minimal surfaces and the topology of three dimensional manifolds with non-negative scalar curvature', Ann. of Math. 110 (1979) 127-142.

4. 'Compact group actions and the topology of manifolds with non-positive curvature', Topology 18 (1979) 361-380.

5. 'On the structure of manifolds with positive scalar curvature', Manuscripta Math. 28 (1979) 159-183.

6. 'On the proof of the positive mass conjecture in general relativity', Comm. Math. Phys. 65 (1979) 45-76.

7. 'Complete three dimensional manifolds with positive Ricci curvature and scalar curvature', Ann. of Math. Studies 102 (1982) 209-228. G.SCHUMACHER

1. 'Harmonic maps of the moduli space of compact Riemann surfaces', Math. Ann. 275 (1986) 455-466. H. C J. SEALEY

1. 'Some properties of harmonic mappings', thesis, University of Warwick, 1980. 2. 'Harmonic maps of small energy', Bull. London Math. Soc. 13 (1981) 405-408 3. 'Some conditions ensuring the vanishing of harmonic differential forms with applications to harmonic maps and Yang-Mills theory', Math. Proc. Camb. Phil. Soc. 91 (1982) 441-452. 4. 'Harmonic diffeomorphisms of surfaces', Harmonic Maps Proc. Tulane, Lecture Notes in Math. 949 (Springer, Berlin, 1982), pp. 140-145. 5. 'The stress-energy tensor and the vanishing of L'2 harmonic forms', preprint. J. SHATAH

1. 'Weak solutions and development of singularities of the SU(2) a-model', Comm. Pure Appl. Math. 41 (1988) 459-469.

C.-L. SEEN

1. 'A generalization of the Schwarz-Ahlfors lemma to the theory of harmonic maps', J. Reine Angew. Math. 348 (1984) 23-33.

204

520

J. EELLS AND L. LEMAIRE

K. SHIBATA

1. 'On the existence of a harmonic mapping', Osaka Math. J. 15 (1963) 173-211. K. SHIGA

1. 'On holomorphic extension from the boundary', Nagoya Math. J. 42 (1971) 57-66. M. SHIFFMAN

1. 'The Plateau problem for minimal surfaces of arbitrary topological structure', Amer. J. Math. 61 (1939) 853-882.

2. 'The Plateau problem for non-relative minima', Ann. of Math. 40 (1939) 834-854. 3. 'Unstable minimal surfaces with several boundaries', Ann. of Math. 43 (1942) 197-222. A. AA SILVEIRA

1. 'Stability of complete noncompact surfaces with constant mean curvature', Math. Ann. 277 (1987) 629-638. L. SIMON

1. 'Asymptotics for a class of non-linear evolution equations, with applications to geometric problems', Ann. of Math. 118 (1983) 525-571.

2. 'Isolated singularities for extrema of geometric variational problems', Proc. Cent. Math. Austr. Nat. Univ. 8 (1984) 46-50.

3. 'Isolated singularities of extrema of geometric variational problems', Harmonic Maps and Minimal Immersions (Montecatini 1984), Lecture Notes in Math. 1161 (Springer, Berlin, 1985), pp. 206-277. L. SIMON and F. R. SMITH

1. 'On the existence of embedded minimal 2-spheres in the 3-sphere, endowed with an arbitrary metric', preprint. J. Si.' o\s

1. 'Minimal varieties in Riemannian manifolds', Ann. of Math. 88 (1968) 62-105. I. M. SINGER

1. 'Infinitesimally homogeneous spaces', Comm. Pure Appl. Math. 13 (1960) 685-697. Y.-T. Siu 1. 'Some remarks on the complex analyticity of harmonic maps', Southeast Asian Bull. Math. 3 (1979) 240-253.

2. 'The complex-analyticity of harmonic maps and the strong rigidity of compact Kahler manifolds', Ann.

of Math. 112(1980)73-Ill. 3. 'Curvature characterization of the hyperquadrics', Duke Math. J. 47 (1980) 641-654. 4. 'Strong rigidity of compact quotients of exceptional bounded symmetric domains', Duke Math. J. 48 (1981) 857-871.

5. 'Complex-analyticity of harmonic maps, vanishing and Lefschetz theorems', J. Dii f. Geom. 17 (1982)

55-138. Y: T. Stu and S: T. YAU 1. 'Complete Kahler manifolds with nonpositive curvature of faster than quadratic decay', Ann. of Math. 105 (1977) 225-264. 2, 'Compact Kahler manifolds of positive bisectional curvature', Invent. Math. 59 (1980) 189-204. S. SMALE

1. 'On the Morse index theorem', J. Math. Mech. 14 (1965) 1049-1055. F. R SMITH

1. 'On the existence of embedded minimal 2-spheres in the 3-sphere, endowed with an arbitrary metric', thesis. University of Melbourne, 1982. R. T. SMITH

1. 'Harmonic mappings of spheres', Amer. J. Math. 97 (1975) 364-385. The second variation formula for harmonic mappings', Proc. Amer. Math. Soc. 47 (1975) 229-236.

2.

B. SMYTH

1. 'Stationary minimal surfaces with boundary on a simplex, Invent. Math. 76 (1984) 411-420. 2. 'The geometry of bounded solutions of the sinh-Gordon equation', preprint. B. SOLOMON

1. 'Harmonic maps to spheres', J. Dii f. Geom. 21 (1985) 151-162. E. SPERNER

1. 'A priori gradient estimates for harmonic mappings', preprint, University of Bonn, 1982. M. STRUWE

1. 'On a free boundary problem for minimal surfaces', Invent. Math. 75 (1984) 547-560. 2. 'On a critical point theory for minimal surfaces spanning a wire in R", J. Reine Angew. Math. 349 (1984) 1-23.

205

ANOTHER REPORT ON HARMONIC MAPS

521

3. 'On the evolution of harmonic mappings of Riemannian surfaces', Comment. Math. Hely. 60 (1985) 558-581.

4. 'Nonuniqueness in the Plateau problem for surfaces of constant mean curvature', Arch. Rat. Mech. Anal. 93 (1986) 135-157. 5. 'A Morse theory for annulus-type minimal surfaces', preprint. 6. 'The existence of surfaces of constant mean curvature with free boundaries', preprint. D. SULLIVAN

1. 'The Dirichlet problem at infinity for a negatively curved manifold', J. Dif, Geom. 18 (1983) 723-732. T. SUNADA

1. 'Rigidity of certain harmonic mappings', Invent. Math. 51 (1979) 297-307. 0. SUZUKI

1. 'Theorems on holomorphic bisectional curvature and pseudoconvexity on Kahler manifolds', Lecture Notes in Math. 798 (Springer, Berlin, 1980), pp. 412-428. 2. 'Pseudoconvexity and holomorphic bisectional curvature on Kahler manifolds', preprint. A. TACHIKAWA

1. 'Rotationally symmetric harmonic maps from a ball into a warped product manifold', Manuscripta Math. 53 (1985) 235-254. 2. 'A regularity theorem for harmonic mappings', J. Reine'Angew. Math. 377 (1987) 1-11. R. TAKAGI and T. TAKAHASHI

1. 'On the principal curvatures of homogeneous hypersurfaces in a sphere', Differential Geometry (in honour of K. Yano) (Kinokuniya, Tokyo, 1972), pp 469-481. K. TAKEGOSHI

1. 'A non-existence theorem for pluriharmonic maps of finite energy', Math. Z. 192 (1986) 21-27. D. TOLEDO

1. 'Bounded harmonic functions on coverings', preprint, University of Utah, 1987. P. TOLKSDORF

1. 'A strong maximum principle and regularity for harmonic mappings', preprint. 2. 'A parametric variational principle for minimal surfaces of varying topological type', J. Reine Angew. Math. 354 (1984) 16-49. 3. 'On minimal surfaces with free boundaries in given homotopy classes', Ann. Inst. H. Poincare, Anal. Non Lin. 2 (1985) 157-165. P. TOMTER

1. 'The spherical Bernstein problem in even dimensions', Bull. Amer. Math. Soc. 11 (1984) 183-185. 2. 'The spherical Bernstein problem in even dimensions and related problems', Acta Math. 158 (1987) 189-212. G. TOTH

1. 'On rigidity of harmonic mappings into spheres', J. London Math. Soc. 26 (1982) 475-486. 2. Harmonic and minimal maps with applications in geometry and physics (E. Horwood, 1984). 3. 'Moduli spaces of harmonic maps between complex projective spaces', preprint, University of Rutgers, 1987.

4. 'On classification of orthogonal multiplication a la do Carmo-Wallach', Geom. Dedicata 22 (1987) 251-254.

5. 'Construction des applications harmoniques non rigides d'un tore dans la sphere', Ann. Global Anal. Geom. 1 (1983) 105-118. A. J. TROMBA

1. 'A new proof that Teichmiiller space is a complex Stein manifold', preprint, Max Planck Inst., 1987. 2. 'Dirichlet's energy and the Nielsen realization problem', preprint, Max Planck Inst., 1987. H. Tsun 1. 'Harmonic maps into noncompact Kahler manifolds', preprint. K.TSUKADA

1. 'Isotropic minimal immersions of spheres into spheres', J. Math. Soc. Japan 35 (1983) 355-379. A. V. TYRIN

1. 'The property of absence of local minima in the multidimensional Dirichlet functional', Uspehi. Max. Nauk 39, no 2 (1984) 193-194. 2. 'Critical points of the multidimensional Dirichlet functional', Mat. Sb. 124 (166) (1984) 146-158 (Russian); English translation, Math. USSR Sb. 52 (1985) no. 1, 141-153. S. UDAGAWA

1. 'Minimal immersions of Kahler manifolds in complex space forms', Tokyo J. Math. 10 (1987) 227-239.

206

J. EELLS AND L. LEMAIRE

522

2. 'Pluriharmonic maps and minimal immersions of Kahler manifolds', J. London Math. Soc. 37 (1988) 375-384. 3. 'Holomorphicity of certain stable harmonic maps and minimal immersions', preprint. K. UHLENBECK

1. 'Regularity for a class of non-linear elliptic systems', Acia Math 138 (1977) 219-240. 2. 'Morse theory by perturbation methods with applications to harmonic maps', Trans. Amer. Math. Soc. 267 (1981) 569-583.

3. 'Closed minimal hypersurfaces in hyperbolic manifolds', Sem. Minimal Submanifolds, Ann. of Math. Studies 103 (E. Bombieri, ed., Princeton University Press, 1983), pp. 147-168. 4. 'Minimal spheres and other conformal variational problems', Sem. Minimal Submanifolds, Ann. of Math. Studies 103 (E. Bombieri, ed., Princeton University Press, 1983), pp. 169-176. 5. 'Harmonic maps into Lie groups (Classical solutions of the chiral model)', preprint, University of Chicago, 1985; revised version, 1988. H. URAKAWA

1. 'Stability of harmonic maps and eigenvalues of the Laplacian', Trans. Amer. Math. Soc. 301 (1987) 557-589.

2. 'Spectral geometry of the second variation operator of harmonic maps', Illinois J. Math., to appear G. VALLI

1. 'On the energy spectrum of harmonic 2-spheres in unitary groups', Topology 27 (1988) 129-136. 2. 'Some remarks on geodesics in gauge groups and harmonic maps', J. Geom. Physics, to appear. 3. 'Harmonic gauges on Riemann surfaces and stable bundles', preprint, University of Warwick, 1987. J.-L. VERDIER

1. 'Two dimensional a-models and harmonic maps from S2 to S2n', Lecture Notes in Phys. 180 (Springer, Berlin, 1982), pp. 136-141.

2. 'Applications harmoniques de S' dans S. I', Birkhduser Math. Series 60, 267-282. J. Vtr MS

1. 'Connections on tangent bundles', J. Diff. Geom. 1 (1967) 235-243. 2. 'Totally geodesic maps', J. Diff. . Geom. 4 (1970) 73-79. W. VON WAHL

1. 'Klassische Losbarkeit im Grossen fur nichtlineare parabolische Systeme and das Verhalten der Losungen fur t-+ co', Nach. Akad. Wiss. Gott. 5 (1981) 131-177. Q.-M. WANG

1. 'Isoparametric functions on Riemannian manifolds I', Math. Ann. 277 (1987) 639-646. 2. 'On totally geodesic spheres in Grassmannians and 0(n)', preprint, Acad. Sinica, 1987. S. M. WEBSt'ER

1. 'Minimal surfaces in a Kahler surface', J. Diff. Geom. 20 (1984) 463-470. S. W. WEI

1. 'Liouville theorems for stable harmonic maps into either strongly unstable, or 6-pinched, manifolds', Proc. Sympos. Pure Math. 44 (1986) 405-412.

2. 'Liouville theorems and regularity of minimizing harmonic maps into super-strongly unstable manifolds', preprint. H. C. WENTE

1. 'Counterexample to a conjecture of H. Hopf', Pacific J. Math. 121 (1986) 193-243. 2. 'Twisted tori with constant mean curvature in R', preprint. B. WHITE

1. 'Existence of least-area mappings of N-dimensional domains', Ann. of Math. 118 (1983) 179-185.

2. 'Regularity of area-minimizing hypersurfaces at boundaries with multiplicity', Sem. Minimal Submanifolds, Ann. of Math. Studies 103 (E. Bombieri, ed., Princeton University Press, 1983), pp. 293-301. 3. 'Mappings that minimize area in their homotopy classes', J. Dii f Geom. 20 (1984) 433-446. 4. 'Homotopy classes in Sobolev spaces and energy minimizing maps', Bull. Amer. Math. Soc. 13 (1985) 166-168.

5. 'Infima of energy functionals in homotopy classes of mappings', J. Dii f. Geom. 23 (1986) 127-142. J. WOLF

1. 'The action of a real semisimple group on a complex flag manifold I: Orbit structure and holomorphic arc components', Bull. Amer. Math. Soc. 75 (1969) 1121-1237. M. WOLF

1. 'The Teichmiiller theory of harmonic maps', thesis, Stanford University, 1986. J. C. WOLFSON

1. 'On minimal surfaces in Kahler manifolds of constant holomorphic sectional curvature', Trans. Amer. Math. Soc. 290 (1985) 627-646.

207

ANOTHER REPORT ON HARMONIC MAPS

523

2. 'Harmonic maps of the two-sphere into the complex hyperquadric', J. Diff. Geom. 24 (1986) 141-152.

3. 'Harmonic sequences and harmonic maps of surfaces into complex Grassman manifolds', J. Diff. Geom. 27 (1988) 161-178. S. WOLPERT

1. 'Non-completeness of the Weil-Petersson metric for Teichmiiller space', Pacific J. Math. 61 (1975) 573-577.

2. 'Chern forms and the Riemann tensor for the moduli space of curves', Invent. Math. 85 (1986) 119-145.

3. 'Geodesic length functions and the Nielsen problem', J Diif. Geom. 25 (1987) 275-296. C. M. WOOD

1. 'Some energy related functionals and their vertical variational theory', thesis, University of Warwick, 1982.

2. 'The Gauss section of a Riemannian immersion', J. London Math. Soc. 33 (1986) 157-168. 3. 'Harmonic sections and Yang-Mills fields', Proc. London. Math. Soc. 54 (1987) 544-558. J. C. WOOD

1. 'Holomorphicity of certain harmonic maps from a surface to complex projective n-space', J. London Math. Soc. (2) 20 (1979) 137-142. 2. 'Non existence of solutions to certain Dirichlet problems for harmonic maps. I', preprint, University of Leeds, 1981.

3. 'Harmonic morphisms, foliations and Gauss maps', Conternp. Math. 49 (1986) 145-184.

4. 'The Gauss map of a harmonic morphism', Differential Geometry, Research Notes in Math. 131 (Pitman, 1985), pp. 149-155.

5. 'Twistor constructions for harmonic maps', Differential Geometry and Differential Equations (Proceedings, Shanghai, 1985), Lecture Notes in Math. 1255 (Springer, Berlin, 1987), pp. 130-159. 6. 'The explicit construction and parametrization of all harmonic maps from the two-sphere to a complex Grassmannian', J. Reine Angew. Math. 386 (1988) 1-31.

7. 'Explicit construction and parametrization of harmonic two-spheres in the unitary group', Proc. London Math Soc., to appear. R. WooD

1. 'A note on polynomial maps of spheres', Invent. Math. 5 (1968) 163-168.

H.H Wu 1. The Bochner technique in differential geometry (E. Horwood, London, 1987). F. O. XAVIER

1. 'The Gauss map of complete non-flat minimal surfaces cannot omit 7 points of the sphere', Ann. of Math. 113 (1981) 211-214. Y. L. XIN

1. 'Some results on stable harmonic maps', Duke Math. J. 47 (1980) 609-613.

2. 'Topology of certain submanifolds in the Euclidean sphere', Proc. Amer. Math. Soc. 82 (1981) 643-648.

3. 'Non-existence and existence for harmonic maps in Riemannian manifolds', Proc. 1981 Shanghai-Hefei Sympos. Diff. Geom. Diff. Eq. (Sci. Press, Beijing, 1984), pp. 529-538 4. 'Regularity of harmonic maps into certain homogeneous spaces', preprint.

5. 'Liouville type theorems and regularity of harmonic maps', Differential Geometry and Differential Equations (Proceedings, Shanghai, 1985), Lecture Notes in Math. 1255 (Springer, Berlin, 1987), pp. 199-208. 6. 'An estimate for the image diameter and its application to submanifolds with parallel mean curvature', Acta Math. Sci 5 (1985) 303-308. 7. 'Stable harmonic maps from complete manifolds', preprint, I.C.T.P., 1986. S. T. YAu

1. 'A general Schwarz lemma for Kahler manifolds', Amer. J. Math. 100 (1978) 197-203.

2. 'Harmonic functions on complete Riemannian manifolds', Comm. Pure Appl. Math. 28 (1975) 201-228.

3. 'Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry', Indiana Univ. Math. J. 25 (1976) 659-670. 4. 'Survey on partial differential equations in differential geometry', Ann. of Math. Studies 102 (1982) 3-71. K. YANG and S. KOBAYASHI

1. 'Prolongations of tensor fields and connections to tangent bundles I: general theory', J. Math. Soc. Japan 18 (1966) 195-210.

208

524

J. EELLS AND L. LEMAIRE

R. YE

1. 'Uber die Existenz, Regularitat and Endlichkeit von Minimalflachen mit freiem Rand', preprint, University of Bonn. 2. 'Regularity of a minimal surface at its free boundary', preprint P. Y. H.

1. 'Quadratic forms between spheres and the non-existence of sums of squares formulae', Math. Proc. Camb. Phil. Soc. 100 (1986) 493-504. Q: H. Yu 1. 'Bounded harmonic maps', Acta Math. Sinica 1 (1985) 16-21. V. E. ZAKHAROV and A. V. MIKHAIIAV

1. 'Relativistically invariant two dimensional models of field theory which are integrable by means of the inverse scattering problem method', Sov. Phys. J.E.T.P. 47 (1978) 1017-1027. V. E. ZAKHAROV and A. B. SHABAT

1. 'Integration of nonlinear equations of mathematical physics by the method of inverse scattering II', Functional Anal. Appl. 13 (1979) 166-174. J.-Q. ZHONG

1. 'The degree of strong nondegeneracy of the bisectional curvature of exceptional bounded symmetric domains', Proc. Internat. Conj. Several Complex Variables Hangzhore (Birkhauser, 1984), pp. 127-140.

Mathematics Institute University of Warwick Coventry CV4 7AL and I.C.T.P. P.O. Box 586 Miramare 34100 Trieste Italy

C.P. 218 Campus Plaine Universite Libre de Bruxelles Bd. du Triomphe 1050 Bruxelles Belgium

209

NOT ANOTHER REPORT ON HARMONIC MAPS J. EELLS AND L. LEMAIRE

In this brief supplement we describe some recent progress on problems raised in the text. We stress that we do not aim at covering all new results in the field. 1. [Jin, Kazdan] have produced a smooth Riemannian metric g on 1R3 and a smooth harmonic map 0: (]R3, g) --, ]R with rank 2 on a half space and rank 3 on its complement. This answers negatively a question of I (3.18) when g is not analytic. 2. Concerning the evolution of maps between compact Riemannian manifolds

by the heat equation (I (6.3) and II (3.40)), [Coron, Ghidagliaj and [Ding] have given examples of explosion in finite time. A large class of such examples follows from this result of [Chen, Ding]: Let (M, g) and (N, h) be compact manifolds with m > 3 where m = dim M and

71 a homotopy class of maps from M to N with inf Ejx = 0. There is an e > 0 such that for ¢o E 71 with E(00) < e, the evolution of 00 explodes in finite time. Quite surprisingly, [Chang, Ding, Ye] have shown that explosion in finite time also occurs for m = 2. In all dimensions m > 2 such examples occur for maps to Euclidean spheres. 3. [Eells, Ferreira] have established the following rendering theorem (1 (11.9), II (5.7)): Let (M, g) and (N, h) be compact manifolds and 71 a homotopy class of maps

from M to N. If m 54 2, then there exists a map 0 E 7i and a smooth metric g` conformally equivalent to g on M such that 0: (M, g") -+ (N, h) is harmonic. The same applies to the Dirichlet problem, when M has a boundary. I (11.9), (12.6) and II (5.8) show that the case m = 2 is completely different, and not yet fully understood for n > 3. 4.

II (3.2)-(3.4) has been completed by [Bethuel]: Let (M, g) and (N, h) be

compact.

If 1 < p < m, then C°°(M, N) is dense in ,CP(M, N) if and only if

7r[p](N) = 0, where [p] is the largest integer smaller than or equal to p. 5. Morrey's regularity theorem for surfaces (1 (10.12), II (3.19)) has been extended by [Helein] to the case of weakly harmonic maps which are not necessarily E-minimising as follows: Let (M, g) and (N, h) be compact manifolds with m = 2 and E ,C1(M, N) a weakly harmonic map. Then ¢ is smooth and therefore harmonic. (Compare II (3.43)). Combining this theorem with 11 (3.46) yields the following generalisation of the removable singularity theorem of Sacks-Uhlenbeck (I (10.15)): If A is a polar subset of M and 0 E G?(M, N) a weakly harmonic map on M \A with in = 2, then 0 is smooth and harmonic on M. On the other hand, the results of Schoen-Uhlenbeck and Giaquinta-Giusti restricting the dimension of the singular set of E-minimising maps in higher dimensions (II (3.21)) will not extend to weakly harmonic maps. Indeed, [Riviere]

210

has announced the existence of weakly harmonic maps 0: D3 _ S2 with 8,0 = D3. 6. There are many harmonic maps from 1R2 to the real hyperbolic plane H2 which have rank 2 almost everywhere. [Choi, Treibergs]. They arise as Gauss maps of certain spacelike surfaces in R1 2 with constant mean curvature. References

F. Bethuel, "Approximation dans des espaces de Sobolev entre deux varietes et groupes d'homotopie", C. R. Acad. Sci. Paris 1, 307 (1988), 293-296. "Densite des fonctions regulieres dans des espaces de Sobolev", These, Univ. de Paris-Sud (1989).

K. C. Chang, W.-Y. Ding and R. Ye, "Finite-time blow-up of the heat flow of harmonic maps from surfaces", J. Diff. Geom., 36 (1992), 507-515. Y.-M. Chen and W.-Y. Ding, "Blow-up and global existence for heat flows of harmonic maps", Invent. Math., 99 (1990), 567-578. H.-I. Choi and A. Treibergs, "Gauss maps of spacelike constant mean curvature hypersurfaces of Minkowski space, J. Diff. Geom., 32 (1990), 775-817. J.-M. Coron and J.-M. Ghidaglia, "Explosion en temps fini pour le flot des applications harmoniques", C. R. Acad. Sci. Paris I, 308 (1989), 339-344. W: Y. Ding, "Blow-up of solutions of heat flows for harmonic maps", Chinese Adv. Math., 19 (1990), 80-92. J. Eells and M. J. Ferreira, "On representing homotopy classes by harmonic maps", Bull. London Math. Soc., 23 (1991), 160-162. F. Helein, "Regularite des applications faiblement harmoniques entre une surface et une variete riemannienne", C. R. Acad. Sci. Paris I, 312 (1991), 591-596. Z. Jin and J. Kazdan, "On the rank of harmonic maps", Math. Z., 207 (1991), 535-537. T. Riviere, "Applications harmoniques de B3 dans S2 partout discontinues", C. R. Acad. Sci. Paris I, 314 (1992), 719-723.

211

INDEX Albanese map, Albanese torus I (7.2), (9.8) Algebraic manifold I (9.7) Almost complex structure I (9.4) II (4.1) (5.23) (7.5) Almost linear function 11(3.13) Andreotti-Frankel conjecture 11(4.29) A priori estimates II (3.12) (12.12) rth associated map I1(7.46) Bernstein problem II (11.83) Birkhoff-Grothendieck theorem 11(4.9) (6.31) (8.18) Blow up I (10. 15) 11 (3,27) (5.3) Boundary 112, 1112 Bounded harmonic function II (11.6) Branched covering 1(9.9) (11.8) Brouwer degree 1(7.4) (11.3) Bubbling off 11(5.3) (5.25)

Canonical lift I1(9.34) q-capacity II (3.45) Casimir operator II (8.58) Category 11(12.33) Cauchy problem II (11.84) Centre of mass 11 (10.49) Cheeger constant 11(11.80) Circle action II (9.42) Clifford algebra 11(10.7) (10.11) Complete lift II (7.12) Complex analytic subvariety 11 (4.17) Complex isotropic map 11(7.24) Complex Laplacian 1(9.23) Complex structure I (9.1) 11 (4.1) Complex variation 11(4.24) (6.29) Composition properties 1(4.1) Condition C 1(6.17) (10.14) Conformal map 1(10.2) Connection I (2.4) Constant function 1I (11.1) Constant map I (3.6) (12.6)11(7.16) (12.4) (12.22) Convex boundary I (12.13)11(12.1) Convex function 1(4.4) I1(2.13) (3.31) Convex supporting domain I (4.4) I1(3.31) Cosymplectic 11(4.3) Coulomb gauge field II (8.45) Covariant derivative 1(2.4) CR map II (4.48) Curvature 1(2.5) Curvature on totally isotropic planes 11 (6.29) (6.45) Curvature operator 11(6.28)

212

Damping condition I (8.7) II (10.28) (10.36) Dehn's lemma I (12.8) II (12.59) Delaunay surfaces I (11.7) Dependence on the boundary values I (12.15) Dependence on the metric II (2.18) Dilatation I (5.9) II (11.49) Dirichlet problem I (12.2) 11 (12.2) (12.78) Disc II (12.4) (12.22)

Eiconal I (8.5) Eigenmap I (8.1) II (10.3) Ellipsoid I1(10.40) Energy I (3.4) II (2.5) (10.48) (10.51) Equivariant map I (4.17) (11.8) II (10.16) (10.24) (12.25) Euler-Lagrange operator I (3.7) (6.19) II (2.8) Existence of harmonic maps I (5.3) (5.4) (6.17) (8.7) (10.9) (10.17) (11.8) (12.5) (12.9) (12.11) I1(3.30) (5.2) (5.6) (10.25) (10.32) (11.20) (11.27) (11.40) (11.85) (12.5) (12.33) (12.50) (12.55) (12.68) (12.72) Extension theorem I (10.15), III (5) Factorisation of maps II (8.18) Finite energy I (5.4) II (11.11) (11.41) Finiteness results I (5.12) (9.13) II (3.55) First fundamental form I (3.1) Flag manifold II (7.65) (9.10) Flat manifold 17 Focal variety II (10.14) Fr6chet curve II (12.44) Fredholm structure II (9.17) Free boundary problem II (12.65)

Gauss-Bonnet estimate I (11.11) Gauss bundle II (8.26) Gauss map I (7.4) (9.7) II (2.35) (10.61) (11.69) Geodesic I (3.9) (6.17) (10.11) Geodesically small disc I (12.9) II (3.14) (1, 1)-geodesic map II (4.43) Graph II (11.75) (12.78) Grassmann bundle II (7.43) Grassmannian I (7.4) II (8.22) (8.32) Group action on a 3-manifold II (12.63) Group action on a torus II (11.22) Growth condition I (5.2) II (11.25) Hamiltonian flow II (9.21) Harmonic coordinates II (3.17) Harmonic diffeomorphism 1 (5.8) (12.14) II (5.10) (11.17) (12.36) Harmonic function I (3.2) (3.6) 11 (11.1) Harmonic map I (3.2) f-harmonic map I (10.20) Harmonic morphism I (4.12) I1(2.30) (7.77) (11.52) Harmonic polynomial I (8.1) I1(10.4) Harmonic reflection 11 (2.38)

213 Harmonic section II (2.39) (8.13) Harmonic sequence 11(8.26) Harmonic variation II (6.21) Hausdorff dimension I1(3.20) (3.27) Heat equation, heat flow 1(6.2) (12.11) II (3.39) (4.30) III (2) Hemisphere II (12.14) Hermitian structure I (9.3) (9.23) Hessian I (3.8) II (2.14) (4.24) 6 (9.44) Holomorphic bisectional curvature I (9.5) (9.14) (9.26) II (4.24) (4.29) (11.54) Holomorphic bundle II (4.6) (4.9) Holomorphic curve II (5.23) Holomorphic function 11 (11.13) Holomorphic map 19 (9.5) (9.11) (12.17) II (4.3) (4.34) (5.23) (11.54) (12.32) (12.37) Holomorphic quadratic differential I (10.5) I1(4.18) Holomorphic sectional curvature 1(9.5) II (4.35) Homomorphism II (8.56) Hopf construction 1(8.4) II (10.5) (10.36) (10.40) Horizontal map II (7.17) (7.28) Horizontally conformal map 1(4.12) II (2.30) Hurwitz's formula I (9.9) q-hyperconvex boundary II (4.32) Identity map II (6.2) Index of a map I (3.8) (3.11) H (2.14) (4.27) (6.9) (6.31) (6.36) (6.48) Instanton II (9.46) Irreducibility condition I1(12.49) Isoparametric hypersurface I (8.5) II (10.8) Isoperimetric inequality 11(11.79) Jacobi field I (3.8) II (2.14) (9.45) Jacobi spectrum II (6.13)

J1,J21I(7.3)(7.27) J-homomorphism II (10.39) J(N) II (7.2) Join I (8.7) I1(10.19) Jordan curve II (12.44)

Kahler angle = e'(0) - e"(O) I (9.17) K91er form 1( 9 . 3 )1 1( 4 .1 )

Kahler structure I (9.5) I1(9.1) (9.12) Koszul-Malgrange structure I1(4.6) (8.23) K(¢) I (9.17)

Gig 1(10.10)11(3.1) Laplacian I (2.16) Lax form II (9.20) Lie group II (8.2) Liouville theorem II (11.6) (11.32) (11.41) Liquid crystal II (12.31) Loop space II 9 Loop theorem II (12.72) Lorentzian manifold 11(7.67) (11.84)

214

2-manifold 110, 11, 11 5 (12.8) (12.32) III (6) 3-manifold II (3.35) (6.23) (12.9) (12.63) 4-manifold II (7.12) Maximum principle I (3.17) II (11.28) (12.10) (12.13) Mean curvature field I (2.21) II (2.21) Mean curvature (constant, parallel) II (2.21) Mean curvature (surface of constant) II (5.26) (11.70) Minimal branched immersion I (10.5) (10.18) (12.7) II (5.17) Minimal embedding I (12.8) II (5.35) (10.53) (12.59) Minimal graph II (11.75) (12.78) Minimal immersion II (2.21) (10.52) Minimal map I (12.7) II (5.17) (12.44) Minimal surface II (5.19) (5.23) (11.59) (12.44) Minima of the energy II (2.5) Minimax procedure II (5.22) (10.31) (11.67) (12.55) Minimising tangent map II (3.27) (11.20) Mixed problem I (12.13) Moment map II (8.45) Monotonicity inequality II (3.22) Morrey's growth lemma II (3.24) Morse index 11 (6.18)

Negative definite of level k tensor II (4.10) Neumann problem I (12.13) II (12.8) Non-compact domain 1( 5 . 4 )1 1 1 1 Non-existence of harmonic map 1(11.6) (11.10) (12.6) II (11.42) (11.85) (12.17) (12.22) (12.27) Non-positive curvature 15 (12.11) II (3.52) Nullity of a map 1 (3.8) 11 (2.14) Orthogonal multiplication I (4.16) (8.4) II (10.5) Partial regularity II (3.21) (3.27) Pendulum equation I (8.9) II (10.23) Penrose programme II (7.19) Perturbed energy 1(6.17) (10.14) II (5.3) Petersson-Weil II (5.56) Pinching theorem 11(6.34) (6.37) Pluriharmonic = (1,1)-geodesic q-plurisubharmonic map II (4.32) (5.56) q-polar set II (3.45) Polyharmonic map I (6.29) Polynomial map 1(8.1) II (10.4) Projective space II (7.47) (8.37) Pseudo-convex domain 11 (12.42)

Quadric bundle II (7.27) Q(N) II (7.27) Rank of a map I (3.18) II (4.38) (4.40) 111 (1) (6) Real isotropic map II (7.7) Reduction theorem I (4.18) II (10.16) Reflection principle I (3.19) (10.8) Regularity 1 (3.3) (3.5) (6.18) (10.12) (12.10) 11 (3.10) (3.21) (3.27) (3.44) (12.3) 111 (5)

215

Rendering problem I (11.9) II (5.7) (10.41) III (3) Riemannian structure I (2.9) Riemannian submersion I (4.7) Riemann surface 1( 1 0 . 4 )1 1( 4 .1 ) Rigidity of Kahler manifolds 11(4.13) Scalar curvature II (6.23) (11.65) Scaling II (3.22) Second fundamental form I (3.1) II (2.8) Second variation I (3.8) II (2.14) (4.24) 6 (9.44) Self duality II (7.4) (7.22) (8.14) Sine-Gordon, Sinh-Gordon equation 11(10.20) Singularities of harmonic maps 1(11.12) (12.10)1I (3.45) (12.21) Small range I (12.9) Space form 11(4.39) Space of maps 1(6.14)1I (3.1)11I (4) Spectral analysis II (6.13) Spectral curve II (8.12) Sphere 18,1110 (12.19) 2-sphere I (10.17) II (5.3) 3-sphere II (8.12) Sphere theorem H (6.33) E-stable, V-stable map II (6.1) (6.26) (6.38) (6.46) (8.8) (11.64) (11.73) Stein manifold I (9.6) II (5.56) (11.16) Stochastic interpretation II (2.43) Stress-energy tensor II (2.25) String 11 (7.67)

Strongly negative curvature II (4.10) (4.40) (5.53) (12.39) Subharmonic function 1(2.18) II (2.13) (4.32) Superminimal II (7.19) Surface 110, 1 1 ,1 1 5 (12.8) (12.32) III (6) Surface of type (p, k) II (12.45) Suspension 1(8.7) II (10.30) Symmetric space II (7.35) (1, 2)-symplectic 11 (4.3)

Tangential Cauchy-Riemann equation I (12.17) II (12.38) Teichmuller theory I (11. 15) 11 (5.46) Tension field I (3.2) Toeplitz operator 11 (9.16) Total absolute curvature II (11.68) Totally geodesic map I (3.1) (3.14) II (2.11) (8.51) (9.41) (10.7) (10.12) Totally geodesic submersion I (4.8) Totally isotropic subspace II (6.30) Twistor degree 11 (7.53) Twistor fibration 11 7 (9.27) (9.41) Umbilic immersion II (7.18) (7.32) Uniformity condition I (12.5) II (11.24) Unique continuation I (3.16) Uniqueness I (5.6) (12.9) II (12.10) Unitary group II (8.15)

216

Variation I (3.7) Variational densities I (6.24) Vector bundle 12 Volume density I (6.27) II (2.21) Warped product I (5.1) (6.13) (10.19) II (3.37) (3.41) Weakly conformal map 1(10.2) Weakly harmonic map 11 (3.7) (3.10) 111 (5) Weierstrass formula II (5.28) (7.61) Weitzenbock formula I (2.17) (9.24) Willmore surfaces II (10.64)

2088 he

ISBN 981-02-1466-9