Harmonic maps

t t=0

= 4>

2

(based on Weitzenbock's method of expressing A ( | d ^ | ) ) established [ l , C h . II] that if M and N are compact and the sectional curvatures of N are nonpositive, then for any initial map <j>' Af - * N the solution ( ) is defined for t

all t > 0 and subconverges to a harmonic map minimizing E and homotopic to <j>. Hartman showed that {t) actually converges [27, §6]; and established uniqueness [27, §5]. Another illuminating proof of the existence of an E-miriimum homotopic to <j>, based on methods of partial regularity, has been given independently by Giaquinta-Giusti and Schoen-Uhlenbeck [28, §3]. Refinements by the latter provide substantially greater applicability. 3.

I n [4] and then [7] homotopy classes of maps between compact Riemann

surfaces were found containing no harmonic representative. T h e holomorphic/conformal structure on the domain was used essentially. For instance, 2

there is no harmonic map of degree ± 1 from a torus T no matter what Riemannian metrics are put on T

2

to the sphere 2

and S . 2

trast, [12] exhibits harmonic maps of all degrees of any T n

projective space CP

2

S,

B y way of coninto the complex

for n > 2.

We have no example of a homotopy class H of maps (j>: ( M , g) —* (N, k) between Riemannian manifolds with d i m M > 3, which does not contain a harmonic map. T h e search for such a class should be reconsidered in terms of the following recent surprise [26]. For any Ti there is a metric g on M conformally equivalent to g and a harmonic map d>; (M, g) —* (N, h) in H, provided d i m M ^ 2. (That exception was expected on elementary grounds, for E is a conformal invariant of surface domains.) 4.

T h e case d i m M = 2 has several other special features which permit

(a) synthetic constructions [7], [11], [12] of harmonic maps of M into manifolds with suitable symmetries; (b) twistor constructions [15], [20], [21], developing a technique in holomorphic geometry initiated by E . Calabi [28, 57]. Nonetheless, our knowledge of harmonic maps with surface domains remains embarrassingly inadequate. 5.

Harmonic maps between Euclidean spheres have resisted general meth-

ods of attack. For instance, the Hopf fibration : S

3

2

—* S

is harmonic,

satisfying the system 2

A $ = \d4>\ 9. (Indeed [27, §8], its components are eigenfunctions of A . ) However, we do not know whether every map S

3

2

—* S

can be deformed to a harmonic map.

A n important contribution was made in the thesis of R . T . Smith [27, §8] and [28, §10]. Working in an equivariant context, he constructed the harmonic join of suitable eigenmaps.

T h a t led to an ordinary differential

equation whose solution - with asymptotic limits - determines an equivariant harmonic map. n

maps S

n

—* S

In that way Smith proved the existence of harmonic

of all degrees, provided n < 7.

W . - Y . Ding [28, §§10.31 and 10.40] recast Smith's reduction as a variational principle for real functions on an interval - thereby permitting somewhat greater flexibility. For instance [25], let n > 3 and Q > , 6) = {far, y) £ R*

+1

x R"

+1

2

2

2

with p 4- q 4- 1 = n and a, b > 0. Assume b ja n

any map d>: Q (a,

n

b) —i Q (a,

2

: \x\ /a

2

+ \y\ /b 2

2

> (n - 3) /4(n

= 1} - 2). T h e n

b) can be deformed to a harmonic map.

Similarly, for any integers k, I there is a harmonic map (in fact, a har3

monic morphism) <j>: Q (a, 2

I

2

b) —* Q (a,

2

2

b) with Hopf invariant k.i iff b /a

2

jk . James

Eells

January

1991

=

1

H A R M O N I C MAPPINGS O F R I E M A N N I A N MANIFOLDS.* By J A M E S E E L L S , J E . and J . H . SAMPSON.

Introduction. With any smooth mapping of one Riemannian manifold into another it is possible to associate a variety of invariantly defined functionals. Each such functional of course determines a class of extremal mappings, in the sense of the calculus of variations, and those extremals, in the very special cases thus far considered, play an important r61e in a number of familiar differential-geometric theories. The present paper is devoted to a rather general study of a functional E of geometrical and physical interest, analogous to energy. Our central problem is that of deforming a given mapping into an extremal of E. Following an infinite-dimensional analogue of the Morse critical point theory, we construct gradient lines of E (in a suitable function space) ; and E is a decreasing function along those lines. With suitable metric and curvature assumptions on the target manifold (assumptions which cannot be entirely circumvented, in view of the examples of §§4E and 10D), we prove that the gradient lines do in fact lead to extremals (see Theorem 11A). If f:M—*M' is a smooth mapping of manifolds whose metrics are gijdx'dx! resp. g $'dy dy , then the energy E(f) is defined by the integral a

&

a

, 9f 9fi a

where the f are local coordinates of the point f(x), *1 being the volume element of M (assumed compact). Thus E(f) can be considered as a generalization of the classical integral of Dirichlet. The Euler-Lagrange equations for E are a system of non-linear partial differential equations of elliptic type:

A is the Laplace-Beltrami operator on M and the r y"* are the Christoffel symbols on M'. Although this system is suggestive of the simple equation Aw4-<£{«) grad w = 0 for one unknown, there is in general very little connection between the two because of the phenomenon of cuTvatuTe. a

2

Received A p r i l 22, 1963. * This work was supported in part by the National Science Foundation and the Office of Naval Research.

109

Reprinted from American Journal of Mathtmatits. V o l L X X X V l , No. 1, January 1964, pp. 109-160.

110

J A M E S E E L L S , J R . AND J . H . SAMPSON.

I t has been necessary to go into the question of existence of solutions in rather great detail, owing to the want of general results for non-linear Bystema. Direct methods of the calculus of variations seem to lead to severe difficulties, and that is one reason why we have preferred to approach the problem through the gradient-line technique, which amounts to replacing the equations above by a system of parabolic equations whose relation to the elliptic system is analogous to that of Fourier's equation to Laplace's equation. This approach is of independent interest, in any case. Our methods are strongly potential-theoretic in nature. The local equations are first replaced by global equations of essentially the same form, embedding i f ' in a Euclidean space. A stability theorem is established showing that a solution of the resulting parabolic system does in fact produce a 1-parameter family of mappings of M into M'. Fundamental solutions of Laplace's equation and the heat equation on (compact) manifolds are used to establish a priori derivatives estimates and to construct solutions of the parabolic system, the latter being translated into a system of non-linear integro-differential equations of the Volterra type, following the method used by MilgTam and Rosenbloom in a linear problem. Curvature enters in a manner not unlike that exploited by Bochner in [4]. Special cases of our extremal mappings go back to the very beginning of differential geometry. E . g . they include geodesies, harmonic functions, and minimal submanifolds. For minimal surfaces they were first studied locally by Bochner [2], in an explicitly Riemannian context. That work was carried to completion by Morrey [19]. In a report prepared by J . H . Sampson in February of 1954 at the Massachusetts Institute of Technology, the subject was taken up from a somewhat different point of view, and other geometrical applications were discussed. Since firm existence proofs were not then available, general publication of the results did not seem warranted. Shortly thereafter, J . Nash and, independently, F . B. Fuller [10] advanced the same definition as that on which this article is based, and Fuller described several examples. The problem has also been considered by E . Rauch. The contents of the present paper are presented in the following order: Chapter I . The concepts of energy and tension. 1. 2. 3. 4. 5.

The energy integral The tension field Invariant formulation Examples The composition of maps

3 HARMONIC MAPPINGS OF RIEMANNIAN MANIFOLDS.

Ill

Chapter I I . Deformations of maps. 6. Deformations by the heat equation 7. Global equations 8. Derivative bounds for the elliptic case 9. Bounds for the parabolic case 10. Successive approximations 11. Harmonic mappings Added in proof: The theory of the energy functional (and its harmonic extremals) is the first-order case of a general theory of y-th order energy (and its polyharnionic extremals). See J . Eells, J r . and J . H . Sampson, L'energie et les deformations en geometrie differentielle, Colloque du CNRS (Proceding of a conference held in Grenoble in July, 1963) for a general formulation in fibre bundles. Chapter I . The concepts of energy and tension. 1.

The energy integral.

(A) Let M and W denote complete Riemannian manifolds of dimensions n and m, respectively, and suppose further that M is closed (i. e., compact and without boundary) and oriented. In the interests of simplicity we assume that both manifolds and their Riemannian metrics are smooth (i.e., of class C°°) ; however, it is not difficult to make minor modification to include differentiability class C . We will let (*%• • denote local coordinates on 31 in a neighborhood of a point P (said to be centered at P if (0. - *,0) are its local coordinates), and (y , ,y ) local coordinates on 31'. Thus we can write the Reimannian metrics g and g' in these local coordinates as 5

1

m = gadx'dxi,

m

ds" = g'apdydyt,

where we observe the summation convention; generally, when using local tensor calculus we follow the notations of Eisenhart [8], We will denote covariant derivatives without the usual commas; e. g., f" t

=

df/dx

l

fff—Pf*/W$$—V^h",

etc.,

where Tif denote the Christoffel symbols. With any smooth map / : 3I—*3I' we assign a real number called its energy, as follows: For each point P £ 31 we let <, >/• denote the inner product on the space of 2-covariant tensors of the tangent space M(P) to 31 at P; thus if (e') Msn is a base for the cotangent space 31'{P) of 3I(P) and a = cttje'®e' and ce" are 2-covariant tensors of - V ( P ) , then l

113

J A M E S E E L L S , J R . A2TD J . H . SAMPSON.

ik

i

where g gk = Z . Since the metric tensor p on M. We will call e(f)P = £ (f*g')(P)>p the energy density of f at P. Its dual differential ra-form e(f) * 1 can then be integrated over M, and with an eye toward the physical concept of kinetic energy (mv /2) we define the energy of the map f by S

i

p

2

£

(1)

(/>=f,

I n a local coordinate representation we have #(f)-i a

f

1

ff'^W^tdettffy)]*^

A • "•A

m

f

where ft" — df /dx . Observe that if the local coordinates centered at P and f(P) are both chosen to be locally Euclidean at their centers, then p

effi ~

I S L W m

so that e(f) is non-negative, and E(f)

vanishes when and only when f is a

constant map. ( B ) Although the present work is devoted primarily to the functional E and its extremals, there will be the indications (e.g. the example § 4 E ) that we will want ultimately to consider other types of energy of maps. We give here a general method of constructing these. Starting with the manifold M and any smooth symmetric 2-covariant tensor field a on M, we fix a point P € M and form the proper values of a relative to the metric tensor g of M- i.e., the n real roots of the equation iet(gn(P)X — O y ( P ) ) = 0 . Apart from their order, these proper values are intrinsically associated with a and P ; thus we are led to forming their symmetric functions. }

Definition. Let a be any symmetric real function of n variables. For any symmetric 2-covariant tensor field a on M we let o-(cr) : M—> R be the function such that
/„(*)= f

f(«)*l. •f:M In particular, let CT„ denote the p-th elementary symmetric function.

s 113

HARMONIC MAPPINGS OF RIEMANNIAN MANIFOLDS.

Then B e t t i n g * , - ! , we have Newton's inequalities /ip-^t,,,! ^ (JI^Y> with equality if and only if all proper values kt are equal; furthermore, if all & S 0, then (2)

ft^W^'^W^S"

' • i f c . p , and

• " ^ * w W i . These provide some sort of comparability of the various o--integrals of a. We have *4$P*0mmtm,


J J

f f l J

(P)],

and in general


2 «l<-<<» • 9<m

Remark. These integrals provide many variational problems of differential geometric interest. For example, if we take for a the Ricci tensor of M, then its proper values are the principal curvatures at a point, and ui(a) is the scalar curvature function. If we take for a a second fundamental form (for a given direction) of an immersion of M, then <7„(a) is the GaussKronecker curvature function (for that direction). For the present purposes we take cc^f*/ for some map /. Then (ft%—f?Wm

and » i ( / V ) — < y » / V > ;

w

e

a r e

l e d

t 0

t

h e

Definition. For any map f: M—*M' and any symmetric function <j of n variables, the u-energy of f is the number

We observe that the energy of f is the (
where the inner product is that of the space of n-vectors of M'(f(P)). volume of f is the number (3)

7(/)=

Note that 7(f) — 0 if 11 > m. 8

f

1

MfV)] *!-

The

114

JAMES B E L L S , J B . AND J . H . SAMPSON. r

Remark. I f we consider g 4- Xf*g = ds\* as a perturbation of the metric of M for A ^ 0, then the corresponding volume element is *lx=[det( /y + A / !

c (

W s)] a

w

A- •

-Adx".

A simple calculation shows that

where V (M) =

f *1>,.

X

(C) We give here an interpretation of E(f) as a measurement of dispersion; the calculations of this paragraph will not be used in any essential way later. Let / ( . f , Q') denote the geodesic distance between the points P', Q'€ M', and suppose P,Q£M are points such that / ( P ) = P ' , f(Q)—Q'. For a fixed P we consider the function Q —* / {P', Q') ; it is elementary that, for Q sufficiently near P, that function is smooth. 2

i!

Let &f = g f{j denote the Laplace-Beltrami operator on the function We calculate the Laplacean of r" qua function of Q. I n coordinates centered at P and P' we have 1

A simple calculation gives 9

y

a

dg dy^ dx* dx> ^ By" *

Now

as Q -> P, whence 2

A / (P',<2')=e(f)P4-0
For any smooth function u defined in a neighborhood of P let u (P) the average value ( * > 0 ) f

f

MP)where B, = {Qe_ M: f{P,Q)

denote

u*l/V(B ), e

^ }. £

We require Maxwell's relation [16, p. 31. The formula is easily proved by expanding u in a Taylor's series at the center of a normal coordinate system.]:

7 HARMONIC MAPPINGS OF RIEMANNIAN MANIFOLDS.

U

UP)

— (P)

115

+ ,/ *u(P)+0(<>). 2

+2)

s

Applying this to u{Q) = / ( P ' , 0 ' ) and noting that « ( P ) = 0 and AtifP) = lim A r " (P', g') — e ( / ) P , 0

we obtain § f*{P',


B

=^e(f)P

+ 0(<*).

If q> (P,Q) is a smooth function which is zero for r(P, Q) ^ t and equals l / F ( S ) for r ( P , Q) then we obtain from our last equation e

t

f

f

v

^(J ,e)* (^0)*l-7ZX^-S(/)+0(«')e

Naturally certain uniformity conditions must be fulfilled by the error terms for the validity of this formula, but since we make no essential use of the result, we shall not insist on that point. The left member of the preceding equation represents the mean square infinitesimal dispersion of the image points on M produced by f, and that quantity is estimated by the energy of f. 2.

The tension field.

(A) I n this section we examine the extremals of E, interpreted as the zeros of the Euler-Lagrange operator associated with E. For this purpose we let 7T: ? (M') -> M' denote the tangent vector bundle. We let 9t (M, M') denote the totality of smooth maps from M to M'; then for every f € 91 (M, M') the set of smooth maps u: M—>3(M') such that i r ° u = j' forms a vector space $i{f) with algebraic operations defined pointwise; such a u is called a vector field along f. We define an inner product in S((f) by f

, *1. IP)

For any vector field v along f the directional derivative of E in the direction v, V , * t f ) - ^ [ £ ( / » ) ] « , where f,(P)

=exp„ >(tv(P)), P

is the endpoint of the geodesic segment in M' starting at f(P) and determined in length and direction by the vector tv(P) € M'(f(P)). We show in ( B ) below that the Euler-Lagrange operator applied to a map f defines a vector field T ( / ) along f which is the contravariant representative of the differential of E at f; i. e.,

8 116

J A M B S E E L L S , J B . AND J . H . SAMPSON. V ^ ( / ) —

< T ( / ) , V > / for

aU

»€#(/).

Thus the maps / for which T ( / ) = 0 are {so to speak) the critical points of E. (B) L E M M A . Let /,: M-^M' be a smooth family of maps for t in some time interval t < t < (j. Then 0

(4)

j E{f ) t

t

- J

M

W

+

g^^yfMwJ-^*^

where A denotes the Laplace-Beltrami operator. Proof.

From the definition (1) we obtain for any

£ > < » - * £ ,

( i , (i) 0

f-nu-a^+w^

f

]«.

The quantities

are the components of a contravariant vector field £ on M, whence by Green's divergence theorem

A routine calculation now gives (4). For any smooth map ft 91{M,M') (5)

we set

r(f)y{P)^Ap(P)+g"(P)r' y(f(P))f "(P)ff(P). as

i

It is clear that r(f)f is unaffected by any transformation of the local coordinates on M near P, and for any such transformation of coordinates near f(P) we see that r{fy*(P) transforms as a contravariant vector in M ' ( f ( P ) ) ; see § 3 for an invariant formulation. Thus with every f£St(M,M') we have a variation r(f) € Si(f). Definition. We call r(f) the tension field of the map f, and we say that / is a harmonic map if ( f ) —0. Thus (4) becomes T

£*<«-- JL*»fc* Suppose now that f is an extremal (with respect to small deformations)

9 -

HARMONIC MAPPINGS OP RIEMANNIAN MANIFOLDS.

117

of E. I f we apply (4) to suitably chosen local deformations (e.g., confined to small geodesic balls on M), we obtain from a standard argument that the Euler-Lagrange equation for the energy functional E is r(f) = 0 . In local coordinates that equation is elliptic. Remark. I t is a simple matter to modify these constructions to include the case that M has a boundary; we should require in Lemma SB that f is a constant function of ( on the boundary. t

2

PROPOSITION. Every map f: M^*M' of class C which satisfies r(f) —0 is smooth. If M and M' are both analytic Riemannian manifolds, then every such map is analytic. The proof of the first assertion is easily obtained by inductive application of [14, Theorem 3, p. 210] ; analyticity foDows directly from [32, p. 4 ] . (C) Our next step is to interpret the tension field r(f) in terms of special coordinate systems on M and M'. For any P € M we let exp : ilf{P) - * M be that map which assigns to each vector « £ M(P) the end point of the geodesic segment emanating from P and determined in length and direction by u; it is elementary that exp is a smooth surjective map, carrying a neighborhood of 0£M(P) diffeomorphically onto a neighborhood V of P in M. If we now choose an orthonormal frame in M(P), then we can use the inverse of exp to refer each point Q £ V to the components relative to that frame. These are called normal coordinates in V centered at P, and they form a coordinate system admissible for the differentiable structure of M. In terms of P-centered normal coordinates we have P

P

P

1)

the metric tensor j y ( P ) = Sy, since the frame is orthonormal;

2)

the Christoffel symbols r ^ / ( P ) = 0 for all i, j , k; that is because the i

equations of the geodesies through P are linear, whence T /'(P)u u i

for

all vectors u6 M{P) ; see Eisenhart [8, p. 54]. 2

i

= o p{P)/b'x dxK LEMMA.

I n particular,

i

= 0 y

fi, (P)

from which we obtain the

For any f€9t (M, M') and point P € M let us fix normal coordi-

nate systems V and V centered at P and P' — fiP),

respectively; in terms

of these we have

Thus f is harmonic if and only if at each pair of points P, f(P) there are such coordinates in which f satisfies the Euclidean-Laplace equation at P.

10 H8

J A M E S E E L L S , J R . AND J . H . SAMPSON.

Example. Suppose M' is fiat; i.e., its Riemannian curvature is zero at every point. I t is well known that then M' admits a smooth coordinate covering such that in each coordinate chart we have T^as" = 0. Then T ( / ) ^ = A/ ", and in such local coordinates the equation r ( f ) — 0 is linear. In particular, if M' is the Euclidean space R , then a map / : M—*R is harmonic if and only if it is constant, by the maximum principle. 1

1

M

M

Example. If M and M' are Lie groups with bi-invariant Riemann metrics, and if / : M —» M' is a homomorphism, then / is harmonic. I t suffices to verify this at the identity of M; taking canonical coordinates (which aTe normal; see Chevalley [5, p. 118]) at the neutral elements of M and M', we see that the representation of f is linear, whence r(f) = 0 by Lemma 2C. Example.

If M and M' are Eahler manifolds and f:M—>Al'

is a

holomorphic map, then f is harmonic relative to the associated real Riemann structures on M and M'. Namely, we take local holomorphic coordinates z> = x> 4- V — 1 s* J and w° = y - -^—ly"'" (where dimJlf — 2h, dim M' — 2k) centered at some points P £ M, f(P) € M'. Because the w" are holomorphic functions of the z> and because M is Kahler, they satisfy Laplace's equation &iv = 0. Now A is a Teal operator, which we have Ay" = 0 = A# " for 1 ^ a <; k. Since M' is Eahler, the w can be chosen to be normal coordinates, whence all r ' « ^ = 0 at f(P). Then ( f ) P = 0. +

a

T

a

i+

a

T

(D)Suppose f:M->M' is a Riemannian immersion. Thus for each P€M the differential f*(P) of f at P maps M{P) isometrically into M'(f(P));i.e.,g = f*g'. It £ \ ( l g o - ^ j r o — n) is an orthonormal frame in (some coordinate system) on M' orthogonal to M, then a

(6)

T

r

)

a

6 l « = ( A i + ' ^ ' ' / i / / ) f f W | = 6„| D

)t

are the components of the second fundamental form 0(f) on M in M' relative to that frame (Eisenhart [8, § 5 0 ] ) . The vector field | along the map f defined by m-n

(?)

iS

HP) =

Hg (P)l> (P)$ (P) IJlij

al

for all P€M is independent of choice of frame. I t is traditionaUy called the mean normal curvature field of the immersion. The following formula is a well known interpretation: m-n HP) SdivU (P))^,(p). 0 |

ir=l

Note that the coefficient of each £ \ is the first elementary symmetric function of the proper values of b \ relative to g; that indicates how the following a

a

11 HARMONIC MAPPINGS OP R I E M A N N I A N MANIFOLDS.

119

results (and those of § 4C) will generalize when we replace energy by o-energy. A Riemannian immersion is said to be a minimal immersion if £ = 0 on M. The minimal immersions are precisely those which are the extremals of the volume functional V (Eisenhart [8, p. 176]). Usually the notion of minimal immersion is taken in a somewhat broader context, as a smooth map which is an extremal of V. There are examples (even for M' = R and n — 3 ) of analytic extremals of V which aTe not strictly immersions, for their Jacobians vanish at isolated points. M

From (7) we conclude that

m-n

y

,

y

a

3

But (fu + r p fi ff )g'vvh'' — 0 ( I g i ^ n ) in general (Eisenhart [8, p. 160]); i.e., r(f) ( P ) is perpendicular to M(P). 0

PROPOSITION. Let f:M—>M' be a Riemannian immersion. Then e(f) =n/2. The tension field r(f) coincides with the mean normal curvature field on f. In particular, f is harmonic if and only if it is minimal. Note that any isometry / : M^M map is harmonic. Remark.

For any « £ M'(f(P))

is harmonic, and that any covering

perpendicular to M(P)

we have

< r ( / ) , « > = trace ( j 8 ( / ) « ) . (E) For any map f £ Si (M, M') we define F: M-*MXM' by F(P) — ( P , f ( P ) ) • then F is a smooth imbedding, but not isometric in the product metric on M X M'< The following result is immediate : PROPOSITION. For any f £ 91 (M,M'} we have e{F) —n/2 4- e(/). Furthermore, f is harmonic if and only if F is harmonic.

3.

Invariant formulation.

(A) I n this section we express our problem in terms of differential forms with values in a vector bundle. I t turns out that in some sense the harmonic theory of such forms gives us an infinitesimal solution to our problem. We begin by summarizing the general theory; for details see Spencer [37] or Bochner [4],

12 120

JA1IEB BBLLB, J H . AND J . H . SAMPSON.

1. Let W—>M be a smooth vector bundle over M with fibre dimension m. If &b''(M)—*M denotes the bundle of p-covectors of M and if W&S'OJJ (M) —* M is the tensor product bundle, then the smooth sections of that latter bundle aTe called the smooth p-forms on M with values in W; their totality forms a vector space, which we denote by Af(M,W). 2. Suppose that W has a Riemannian structure; i. e., a given reduction of its structural group to the orthogonal group in dimension m. If W -> W*. denotes the dual bundle of VP, then we have a bundle isomorphism \ji: W —* W*. Taken together with the canonical isomorphism *: 3 [M) —> 3 ^{M) induced from the given Riemannian structure on M, we have the natural map l v i

In particular, the evaluation of

A*(M,W)->A*-P(M,W*).

inr

W*

on

W induces a bilinear pairing #:

xAP{M,W)^A"(M),

A*(M,W)

the vector space of real valued ft-forms on M . Thus A " ( M , W) has the inner product (8)

f

<<M> =

*#f-

J M

3. Assume next that TP has a connection which is compatible with its Riemannian structure; i. e., the covariant diSerential of the tensor field defining the orthogonal reduction of W is zero. We say that W is a Riemannian-connected bundle. Then we have a linear map d: AP(M, W) —> A ^ ^ M , W), which can be described as follows: I f % is a locally finite covering of M by coordinate systems, then for each V 6 1l the connection in W can be given by a certain m X m matrix 8 = (9a &) of 1-forms in U. Similarly any form di € A ? ( M , W) defines an m-tuple <j> = (^>p ) of j-forms in P. If d denotes the exterior differential operator in TJ, then we define d

u

u

v

u

a

(dd>) = dv + 6U A4>

4. We can now develop the theory of harmonic forms on M with values in a Riemannian-connected bundle. We have the adjoint map 8: A"(M, W) —* A ^ (M, W), characterized by the formula J

1

13 HAEMONIO MAPPINGS OF R I E M A N N I A N MANIFOLDS.

121

1

I n particular, for any
v a

f l ^ a ^ m ) , which can be expressed u

u

n

in the coordinates

• • ,x )

a

in U.

If we write similarly

then

(9) {

letting V i denote covariant differentiation on M relative x . The Dirichlet integral of * £ A"{M, W) is 2ZJ(d>) — > 4- <Sd>, and its associated Laplace operator is A =—((i84-S<2). A form d>£ A*(M, W) is harmonic if Atp^O; that condition is equivalent to the pair d
A»(M,W)

=H"{M,W)

© [dA^(M,W)

1

+

&A'» (M,W)~\

I

P

where H (M, W), the space of harmonic p-forms with values in W, is orthogonal to the other two summands. 5. The curvature of the connection can be expressed in U by an m X W matrix of 2-forms ® = ( ® ) where ® = dd + 0 A^ I f V denotes the covariant differential of M, then we define the covariant differential V on 4>eA*(M,W) by u

u

u

u

U

u

a f t

Letting i i / denote Ri"I, the diagonal m X m matrix each of whose diagonal terms is the Eicci tensor field Ri" (we follow the sign convensions in Eisenhart in defining By—-jBPyi)., we have for any q> £ A (M, W) the following expression for the components of the Laplacean of

where ( denotes the transposition of matrices. 1

I f tb^^ A (M,W), define the function Z7 we have (usual Laplacean A of functions)

*##f)

on M ; then in

14 122

J A M E S E E L L S , JR. AND J . H . SAMPSON. u

where a denotes the Riemannian structure of W in V. For each

W)

define the function m)= a® ) -m('P u m*.

(id

,

u

i

h

a

v

The matrix (of functions) of Q is

We consider this as an nm X rim matrix in the subscripts (ah), (0i); as such, it is symmetric: QaB — Qfx* The integral over M of &(d>-di/2) is always zero, by Green's theorem. Thus if
(12)

ih

f
f

( V ^ ^ r f ^ V ^ ' t ^ J M ^ O .

( B ) Given any / £ 9* (M, M'), let /"'ff (M') - » M be the induced vector bundle; it is clearly Riemannian-connected. Let us interpret the preceding development for that bundle. l

First of all, the elements of A°(M,f- 3(M')) are canontcally identified with the vector fields along f (i.e., with the elements of the space 91(f), in the notation of § 2 A ) . Secondly, for any PtM the differential f * ( P ) : M(P)—>M'(f(P)) is a linear map, to be considered as an element of M'(f(P))®MW(P); otherwise said, the assignment P^>f*(P) determines a specific 1-form / * £ A (M,f- -3 (W)). Thirdly, we have 1

,

J M

LEMMA. For any /£ Sf(Sl,M') tolA*(M,f-^3(M')). Thus if* Proof.

we have Sf* = 0; i.e., f„ is orthogonal dUf*.

Take a coordinate chart U on M, and write

(13)

SyW^T'viPffdxK u

Then {U )* — fffc>, (

^

D

)

whence 7

" w

(l
A

m

+ v w * *

a

m

which is zero, because both coefficients are symmetric in t, j . r

Similarly, the variation gf £ A°(M, f'^3 (M )) tation +

&f*h

u

s^V.Vi/HrVfW)

has coordinate represen4 m

15 HARMONIC MAPPINGS OP R I E M A N N I A N MANIFOLDS.

123

PROPOSITION. For any f € St (M, M') its differential f#is a closed 1-form. Its tension field r(f) = — 8/*, the divergence of its differential. The map f is harmonic if and only if its differential is a harmonic 1-form. Definition. For any map f S Si (M, M') its fundamental form f3(f) is the covariant differential V / * of its differential. Thus fi(f) is the f- 3(M')vatued 2-covariant tensor field on 31 whose coordinate representation is 1

m -

B

Sij-'+^fi% -f

^-

The tension field r(f) is just the trace of fi(f); i.e., &(f\—g*§n^ ( l ^ T ^ m ) . I t follows from § 2D that if f is a Riemannian immersion, then 0(f) is the second fundamental form of M in 31'. Analogously, let us say that a map / € 9i(M,M') is totally geodesic if 0(f) —0 on M; we will see as a consequence of Corollary 5A below that totally geodesic maps map geodesies into geodesies. u

(C) Let us consider the function Q(f*); from the expression (® 0 )i* = g R'aBi$fk"fi we compute (11), taking into account the skew symmetry ® ®ea to obtain the a

Rk

u

S

u

oB

LEMMA. (14) Its

For any smooth map f: M-*M' we have Q(U)

W

W

W

1

'

1

matrix (for arbitrary forms d>£ A (M,f- 3(M')))

(15)

-

W

*

is

7

Q «

PW* ftW'—

aB

If f is harmonic, then (16)

*e(f) = \J3(f)\* + Q(f*)-

We will refer to the matrix (15) as the Ricci transformation on the tensor product bundle f~*3 (M') ® 3 (M). Observe that if / is a real-valued function on M, then that Ricci transformation is just that given by the Ricci tensor of M. [l]

Remark. The above computations can of course be made without passing through the medium of vector-bundle-valued differential forms. One starts by applying the Ricci identities (Eisenhart [8, p. 30]) to the direct evaluation of &e(f), and then reads off the appropriate terms. The next result follows the well known pattern of Bochner; in [2] Bochner has also applied the method in a special case for maps. THEOREM.

If f: M->M' is a harmonic map, then

16 124

J A M E S B E L L S , J R . AND J . H . SAMPSOV.

r

«(/*)*!

m

J M

and equality holds when and only when f is totally geodesic. Furthermore, *f Q{f*)=® 3fj then f is totally geodesic and has constant energy density e(f). o

Proof.

n

This follows at once from Stokes' Theorem f

Ae(f)*l = 0,

applied to (16); for if Ae(f) £ £ 9 , then A e ( f ) = 0 everywhere, whence e{f) is a constant function. Following the conventions of Eisenhart [8], we say that the Ricci curvature of M is non-negative if at every point P £ M the matrix (—R (P) ) is positive semi-definite. tf

COROLLARY. Suppose that the Ricci curvature of M is non-negative and that the Riemannian curvature of M' is non~positive. Then a map f: M—*M' is harmonic if and only if it is totally geodesic. Furthermore, 1) if there is at least one point of M at which its Ricci curvature is positive, then every harmonic map f: M—*M' is constant; 2) if the Riemannian curvature of M' is everywhere negative, then every harmonic map f:M-*M' is either constant or maps M onto a closed geodesic of M'. Proof.

The theorem shows that

If hypothesis 1) is satisfied at ? £ M, then / * { P ) = 0 , whence the constant ( / ) = 0 ; i- v / is a constant map. If hypothesis 2) is satisfied at P = f(P)£M' and we take normal coordinates centered at P, then the f*(P)-image of the tangent space M(P) has dimension £ 1 . If it has dimension 0 at any f(P), then again e(f) — 0; otherwise, the image f (P)M(P) has constant dimension 1. Because f is totally geodesic, the conclusion follows. e

e

r

tl

Example. If f: M—*M' is a harmonic immersion, then e ( / ) = n / 2 , and | j3{f)\ -\- Q(f#) = 0. This relation also follows from Gauss's equations (Eisenhart [8, p. 162]) 2

17 HARMONIC MAPPINGS OP RIEMANNIAN MANIFOLDS. 11

by multiplying by g^g

and summing.

135

We obtain n-ni ir=I

il

where fi = g h \i is the w-th the immersion (see ( 7 ) ) ; each Riemannian curvature of M' R — R gy of M is negative immersion of M in M'. a

a

j

iS

component of the mean normal curvature of p.„ — 0 if / is harmonic. FOT instance, if the is non-positive and if the scalar curvature at some point, then there is no harmonic

4. Examples. 1

(A) The case dim M = 1. Let us take for M the unit cirele S , coordin a t e d by the central angle 6. For any f£(S\M') we have 2 *1.

The tension field is t ( / )

+

W -"*

dt

df'

which (when the parameter of f is proportional to arc length) is often called the curvature (or acceleration) of f.

We have = 0, and -r(f) is dt proportional to the geodesic curvature vector field along /. Then f is harmonic if and only if f defines a closed geodesic on M'. It is well known that if M' is compact, then in every homotopy class of maps S —* M' there is a harmonic map (and furthermore, one which minimizes the length in that homotopy class.) On the other hand, without further restrictions on g" there are complete Eiemannian manifolds M' and non-trivial homotopy classes of maps 8 —> M' having no harmonic representatives; see § 10 below. 1

1

(B) The case dim M = 2. We established here certain relations showing the close connection of our problem with the Plateau problem, in its potential theoretic formulation (Morrey [ 1 9 ] and Bochner [ 3 ] } ; incidentally, we see that in our energy theory the cases dim M £j 2 are favored. Recall that a map h: M->M' is conformat if there is a smooth function 6: M^»R such that h*g' = exrj(2e)g. Thus the differential h% preserves orthogonality and dilatates uniformly. Clearly such a map is a smooth immersion, and has energy density e(h) =

7iexp(20)/S.

18 126

J A M E S EET.T.S, J B . AND J . H . SAMPSON.

PHOPOBITION. If dimM = w = 2 and h: M-*M is a conformal difeomorphism, then for all f€3t(M,M') we have E(foh) = E(f). Moreover, h is harmonic. a

tl

a

3

Proof. First of all, (/ ° k), = f «hf, whence 3e(f °h) = g hfhfif f g' . The conformaltity condition foT h implies g'thfhj" = exp(%0)gM, and substituting gives f

3e

(f

f

o h) - ex (Z$)g^Wf/g' V

II

aB

- 3 exp(20) e(/).

aB

Secondly, we have -

[det(j Vty)/det( ,,)]»*l F t

f f

— esp(n#)*l, so that if 7i = 3 we have h*(e{f)*l)

= e(f"°ft)*l.

Finally, in suitable local coordinates on M we have iy = i v and

direct computation shows that T

V(f)

+V*,—y*»y
(ft)* = (2 — n)<7% .

(l^i^n).

PROPOSITION, dimilf— 2, then for any f€Si(M,M') we have ^E(f). Equality holds when and only when f is conformal.

Proof. The first statement follows immediately from the inequalities (2). Suppose f is conformal; then for fi = 2 we obtain y(f)-

f

exp(2t?}[ „^?

ConverBelyj if V(f) —E{f)

2 5 l 2

]S(2^=

(

e{f)*l =

E(f).

we conclude that at every point of M 1

tJ

2[det(/V)y] = ? (/*?')«[det(^)]l. In isothermal coordinates on M we have LW)n-(f*g'UY

{ ( r Y ^ n

whence (f*g')u— tf*9')*2 — 0 = ( / * / ) » • exp(2fl) -

Defining 6:M-*R

by

[det(/*/) /d t(^)]i, y

e

we obtain exp(2fi)jy = (f*g')nCOEOLLABY. f minimizes E. 0

If a map f„: M—>M' minimizes V and is conformal, then

19 HARMONIC MAPPINGS OF RIEMANNIAN MANIFOLDS.

127

For any f£ Si (M, M') we have E(f )=V(f )^V(f)^E(f). 0

0

Remark, I f M is a Riemann surface, then its complex structure defines a conformal equivalence class of Riemann structures. The energy of any map /: M—*M' therefore depends only on the complex structure of M. (C) Harmonic fibre maps. For any map f £ Si{M, W) and P€ M we have the vector space Mp(P) = ( u £ M(P): / ( P ) j t = 0}; the vectors in Mv(P) are called vertical. Let J V H ( P ) denote its orthogonal complement in M(P). Suppose that for all P e M the differential f*(P) maps M (P) isometrically onto M'(f(P)). Then / is a locaUy trivial surjective fibre map (see Hermann [13]) ; in particular, / determines an almost product structure on M (i. e., the structural group 0„ of 3 (M) admits a reduction to the product group O X On-m). We will call a map / : M^>M' a Riemannian fibration. +

H

m

Remark. There are smooth titrations / : M—*M? having no Lie structural group; e. g., there are non-trivial compact smooth fibrations over the 3-sphere S" (which cannot have a Lie structural group G, since ir ((?) = 0 ) . I t is a consequence of a theorem of Hermann [13] that the manifolds M, M' admit no Riemann structures compatible with f as above, for which the fibres are totally geodesic. 2

L E M M A . If f: M-*M' is a Riemannian fibration, then for any P e l f there are coordinate charts U and V centered at P and f(P) respectively, in terms of which fi (P) = &V" (1 = i = , 1 = = fft). Furthermore, the first m coordinates in U can be considered as normal coordinates in V, and the last n —m coordinates are local coordinates for the fibres. a

n

a

Proof. Let ( e ) i s „ be an orthonormal base for M(P) such that the first m vectors span MH(P), and the last n — m vectors span My(P). Then f#(P)ei = e\ (lt=i = m) form an orthonorrnal base for M'(f(P)), and we can construct the associated normal coordinates in some neighborhood V. According to Hermann [13] the unique horizontal lift to P of any geodesic of M' starting at f(P) is a geodesic of M; one determined by / lifts to one determined by e . We now use the local product structure to define a coordinate chart V in which the fibres have the desired property. Note that (unless the fibres are totally geodesic) we cannot generally require that the coordinates in U be normal. 1

5

(

(

PROPOSITION.

Let f: M->M'

be a Riemannian fibration. Then e{f)

20 128

J A M E S E E L L S , J B . AND J . H . SAMPSON.

= n/2. If for any P€M we let F denote the fibre through P and i : the inclusion map, then r(f)(Q) U(Q)r(i )(Q) for all QtF . P

P

P

Fp-*M

P

In particular, f is harmonic if and only if all fibres are minimal submanifolds. Proof. In § 2D we have seen that the tension field of any Riemannian immersion is perpendicular to the submanifold. Thus for every Q£ F we have r(i ) (Q) e M (Q). I f we use a split coordinate system as in the previous lemma, we see that m UT(i )(P)t=-2h->Hir) [P)*=i P

P

U

k

P

The proposition foUows by direct calculation. Examples. All covering maps are harmonic; in particular, the identity map is harmonic, which amounts to saying that Cartan's vitesse is a harmonic 1-form. If 3 (M)^*M is the bundle of orthonormal r-frames of M, then it is known (Liehnerowiez [15]) that with its natural Riemannian structure on 3 (M) the fibres are minimal, whence that fibre map is harmonic. Vector bundle maps are harmonic. Every homogeneous Riemannian fibration is harmonic, for the fibres are always totally geodesic, and therefore minimal. r

T

1

(D) Maps into flat manifolds. Let us take for M' the unit circle S ; we will construct harmonic representatives in every homotopy class of maps M—>S . First of all, it is well known that the set [M, S ] of these homotopy classes forms an abelian group canonically isomorphic to the first integral cohomology group 2f*(jflfJ. Secondly, every such cohomology class is canonically represented by a harmonic 1-form on M. l

1

Suppose M is connected, and fix a point P e M. Given any such harmonic 1-form to on M and any smooth path y from P to a point P e M, we define the number B

P

0

tm -1* A different choice y of yp may give a different number f(P), but P

Ki>)-/(P)=

f

«

is an integer since the periods of a are integral; consequently, oi determines a well defined map f : it-*. S by letting f (P) be the residue class modulo 1 of f(P). 1

a

a

21 HARMONIC MAPPINGS OF R I E M A N N LAN MANIFOLDS.

129

Now since f is harmonic, every P e M has a neighborhood U in which df = m; thus A/ = Sdf 4- dSf = Sm = 0 in f7. I t is easy to see that n—^f establishes an isomorphism [M,S ]=H (M). a

a

1

1

To define harmonic representatives of the homotopy classes of maps M—> T , the flat m-torus, we merely take m-fold products of harmonic maps M—*S*. using § 5 C below; the existence of these harmonic representatives was first proved by F . B. Fuller [10]. More generally, any compact flat manifold M' is covered by T , by a theorem of Bieberbach, and any homotopy class of maps M—>M' which can be lifted to maps M—>T has harmonic representatives obtained by composition with the projection T —>M'. m

m

m

m

I f M and M' are both flat, then the only harmonic maps M-*M' are those which are locally linear, as can be seen from the maximum principle. ( E ) Maps of Euclidean spheres. Let S"(r) denote the Euclidean nsphere of radius r; write S" = S" (1). Then the homotopy classes of maps of S" into itself are classified by their degrees. We consider now the problem of constructing explicitly harmonic maps of a given degree fc; it is a simple matter to modify the following remarks to include the case of maps yS"(r)->S"(r'). 1

1

n

If (x ,- • •.a;"* ) are Euclidean coordinates and JJ: S (r) —>$"(/) is a map given by (jrfaV " " . E " * ) = ( z V / j y - - , i t f ^ V / r ) , then 1

6

{

y

i

=

)

n

/

{ry.

W

We will henceforth refer points of S" to coordinates (#,<£), where 6 denotes colatitude ( O ^ 0 S = T T ) , and tj> a point on the equator r?" of S*. Furthermore, corresponding to the integer Tc let * : S"' (r) —* fl* *^!*) be the (n — 3)fold suspension of the map ^ ( r ) —»S (r ) defined by —* (cos k$, sin h
1

1

(17)

*W =

{

f

c

Z

+

3

1

/

W

~

3

>

(

/

A

>

i

-

n

We are interested in maps f: S»-*8 ol the form (9,^) - » ( © , # ) , where © is a function of 0 alone, and * is defined for r = j = 1. From ( 1 7 ) we see that on ^ ( s m f l ) we have 2e(f) = (fc + n — 3 ) s i n ® / s i n 0 , whence /

2

(18)

9

s

I

22 130

JAMES E E L L S , J R . AND J . EC. SAMPSON.

I n case n = 1 the existence and properties of harmonic maps of degree k is elementary. We now consider the case n = 3. We have gn = l, <7i2 = 0 = 3 i, g = sin 0. The tension field of any map f is given by its components a

2

2Z

(Jf)"],

=Ae-sin cos«[^(g) + Z

T W

0

, „

;

r

0* d® ,

1

a* 0®-]

For the special maps under consideration we find that for a harmonic map f r ( * ) = A* — d^/d

1

d

(

•

d a

®\

>! sinOcos®

the only solution of T ( @ ) = 0 regular at the poles is =k

(19)

® — 3arctan [c(t&n8/%) '] with c > 0.

Then = ± ksin®/sin#, from which we can = 4TT I k | ; note that it is independent of c. Finally, for the degree of a map shows that degree f = ± & . k = 0 and all c > 0 the map is constant; for k — ± 1 the identity and the antipodal map, respectively.

conclude that E(f) the integral formula We observe that for and c = 1 the map is

Remark. Although the above construction does exhibit a harmonic map in every homotopy class, it does not begin to exhaust their topological interest. For instance, with the uniform topology on $t(S ,S ), the component 9t ,(S ,8 ) of those maps of degree 0 has infinite cyclic homology: 2

1

2

s

<

H ^ S l ^ , * ) ) — S

2

) )

=

W 3

(S^)=Z,

generated by the Hopf map. A harmonic representative of that generator should have positive Morse index. Consider now the case n 3 3. We do not know how to construct harmonic maps of degree 3. Incidentally, the suspensions of the above maps (and their "compressed" suspensions below) are generally not harmonic. We propose now to show that the functional E: 91 (S , S") - » R does not have an absolute minimum on the component 9l (S"' 8") for k^O and « > 3; that this phenonmenon could be illustrated explicitly and simply by maps of spheres was suggested to us by C. B. Morrey. : : >

x

k

!

Let &(»><#)!•— ( » 4 * ) , « ) be the map of S»-*S*

defined using (19)

23 HAEMONIC MAPPINGS OP BIEMANNIAN MANIFOLDS,

131

with exponent fc ^ 0. Thus for small c > 0 the map /„ compresses most of S" into a small cell centered at the pole, and that compression takes place along longitudes. The energy integral (18) now takes the form

2

Jo

V, amff

/

For » £ 3 w e have

The following lemma shows that 3 i m £ ( f ) = 0 ; i.e., for n S 3 there are c

maps in Sl (8",8 ) map f £ 9t (8 ,S ) has degree 0. n

k

n

n

k

LEMMA. J

of arhitrarily small energy. But for fe^O there is no with zero energy, for such an f is constant, and therefore

2

Bm ® (6)d6^> e

0 as e - » 0.

Proof. For any < > 0 let a = —1/2. There is a number K such that 0 ^ (tan 0/2) g-BTfor all 0 ^ 0 £ p. I t follows that 0 , ( 0 ) ^ 2 arctan(eff), whence there is a number c > 0 for which 0 s i n 0 ( 0 ) < t/2p if 0 < e :£ c . Thus 2

r

c

*/p

5, (A)

«/o

f

-*

p

The composition of maps.

The following computation is elementary.

L E M M A . If / : M—*M' and f: M'-*M" their fundamental forms satisfy (20)

( f » /)

- ^ f V +

are any smooth maps, then

f ^fm ;

COBOLLABY. TAe composition of totally geodesic maps is totally geodesic. The inverse of a totally geodesic diffeomorphism is totally geodesic. COHOLLABY.

(21)

r(fof)^ (f)^ T

+

fifrfffif

If f is harmonic and f totally geodesic, then f of is harmonic. I n general, however, we do not expect the composition of harmonic maps to be harmonic, as the following example shows:

24

133

JAMES E E L L S , JR- AND J . H . SAMPSON. 2

Example. Let T be the flat 3-torus parametrized by the angles (6, <j>) with 0 g 0, S be defined by 2

3

f(6,4>) — {cosfl, s i n 0 , c o s ^ , s i n ^ ) / V 2

;

2

considered as a point in fi*. Then / ' defines a Riemannian imbedding of T in S , which is a minimal hut not a totally geodesic imbedding. To see that f is harmonic we show that T ( / ' ) is perpendicular to S ( / ' ( P ) ) in R*, and then appeal to Proposition 5 B below. Namely, because T is flat we have 3

3

2

W

=

W

+

£

—

1

f

*

l

=

a

4

=

)

3

whence r(f)(P) is directed along the radius of S at f(P). On the other hand, T is not totally geodesic in 8 , for the map / : S -*T defined by /(6) = (0,0) is a geodesic of T ; it does not lie in any 2-plane through 0 in R*, and is therefore not a geodesic of 3 . I n particular, / : S —>T and f: T —*S are both harmonic maps, and their composition f'°f is not. 2

3

l

2

2

s

2

l

2

3

(B) PROPOSITION. / / f: M'-*M" is a Riemannian immersion, then for any map f: M-*M' we have E(f) =E(f°f). The tension field r(f) is the projection on M' of the tension field -r(f°f). Proof.

The first statement follows from the equation o

«(f /)-t-«(/)The second statement is a consequence of (31); for if f is a Riemannian immersion, then the right member is the decomposition of r(f°f) into horizontal and vertical components because {f ,ap") is the second fundamental form of /'. COROLLARY. A map f: M—*M' is harmonic if and only if r(fof) perpendicular to M'{f(P)) for all P€M.

is

This generalizes the classical fact that a curve in M' is a geodesic if and only if its curvature vector in M" is always perpendicular to M'. ( C ) PROPOSITION. Let f:M'->M" be a Riemannian reducible fibration with totally geodesic fibres. Then for any map f: M—*M' we have Hf°n=r*(Hf))This is immediate, because we can in the present situation take split normal coordinates in Lemma 4 C .

25 HARMONIC MAPPINGS OF RIEMANNIAN MANIFOLDS.

133

COROLLAEY. If M = M" and f: M-*M' is a section, then f ° / = l is harmonic, and therefore T ( / ) is always vertical. Example. If we view a smooth r-form u> on M' as a section of the bundle 3 (M') of r-covectors of M', then the condition that ui be harmonic in the sense of de Rham-Hodge is generally different from the condition T(W) = 0 , rising (say) the Riemannian structure on S (M') of Sasaki [ 2 5 ] . However, these two concepts do coincide if M' is flat. lrl

[r]

Example. A map / : M M' X M" into a Riemannian product has a canonical decomposition f(P) = ( f (P), f (P)) for all PeM. Then / is harmonic if and only if both components /', f are harmonic. For instance, seee Proposition 2B. (D) Let us suppose that M' is a Riemannian submanifold of M" and that the imbedding is proper; i. e„ such that the inverse image of any compact subset of M" is compact in M'. Since M' is complete, there is a positive smooth function p : M' —> R such that for any P' £ M' the set {P"el£":

J

r {P\p")

gp(P')}

is geodesically convex in M"; if M' is compact, then of course we can suppose that p is a positive constant. For each P' £ M' let D - denote the closed ball of dimension q — m (q = dim M") consisting of all geodesic segments of length ^p(P') emanating from P' and perpendicular to M'(P'). The following result is well known and elementary. P

r

L E M M A . Let iV = U {Dp-: f € M ); then N is a neighborhood of M' in M", and the obvious map w: N —* M' defines a smooth fibre bundle over M' whose structure group is 0 . and whose fibres are closed balls. Q m

Taking into account Proposition 5 C we have the PROPOSITION. Let f: M'—*M" be a proper Riemannian imbedding and tr: N—>M' a normal tubular neighborhood. Then for any map f:M—*N the composition •wof is harmonic if and only if r(f) is vertical.

Chapter I I . Deformations of Maps. 6.

Deformations by the heat equation.

(A) This chapter is devoted to the fundamental problem of deforming a given map into a harmonic map; i.e., into a smooth map f:M-*M' satisfying the nonlinear elliptic equation

26 134

J A M E S E E L L S , J R . AND J . H . SAMPSON.

(1)

T(/)-0.

We begin by discussing general methods of attack. The interpretation given in § 2A of the tension field T ( / ) as the contravariant representative of the differential of E at / suggests that we try to invent gradient lines of E in a suitable function space of maps from M to if', and then to prove that these trajectories lead to critical points of E. We propose the following method for realizing such an attempt, which we now outline briefly. We do not pursue this method in the present paper, although the qualitative results are essentially those of the following sections. r

Let 9l {M,M') denote the function space of all maps from M to M' whose partial derivatives (relative to fixed coordinate coverings) of orders ^ T are square integrable. An inequality of Sobolev insures that if 2r > dim M then the maps in $l (M,M') are continuous, and its topology is larger than the uniform topology. It can be shown that the space 3l (M, M') admits an infinite dimensional Riemannian manifold structure modeled on a separable Hilbert space, and that E: 9t (M,M')-»R is a differentiable function. I f r is its gradient field on 9t {M,M'); i.e., V«E(f) < T (/)>»> for aU vectors v in the tangent space at /, then the ordinary differential equation r

r

r

f

T

t

has a local solution which is unique; furthermore, E(f ) is a decreasing function of t. Under suitable curvature restrictions the solutions are globally defined. I f each trajectory / is relatively compact, then it haB a limit point a harmonic map. Thus these trajectories define a canonical homotopy of the initial map onto a harmonic map; moreover, such trajectories enjoy the 1parameter group property. We observe that the critical points of E are just the zeros of all r (for any T > dim M / 2 ) . t

(

r

a

Now the function space 9i {M,M') is not a manifold, although with every map / we have the Hilbert space 9t°{f) of vector fields along / defined i n § 2 A . In particular, ° ( / ) — r ( / ) is in # " ( / ) . I n analogy with the above outline we are led to consider the nonlinear parabolic equation T

(2)

If—**/*}

((o<(
The study of this equation is our primary object in the foUowing sections. We will find that the properties of the trajectories of (2) include most of those mentioned as belonging to the trajectories of t* (2r > dimM). (There is one basic difference: The solutions of (2) are generally defined only for

HARMONIC MAPPINGS OF RIEMANNIAN MANIFOLDS.

135

r

non-negative time, whereas the trajectories of r are always defined for an open time interval around i = 0.) Remark. I n the calculus of variations a standard method (Morrey [ 1 9 ] ) of establishing the existence of a minimum of E for a given class of maps is to take the space M (M,M') and to introduce on it a weak topology relative to which 1 ) E is lower semi-continuous, and 2) there are sufficiently many compact sets. That approach works well for dim i f = 1 or 2; however, the example given in § 4 E shows that it will not work in general for n S 3. 1

(B) PROPOSITION. If ( i , P ) - » f , ( P ) is a map of (t ,t,)XM^M' which is C on the product manifold and C on M for each t, and if that map satisfies (2), then it is C. 0

1

!

We will refer to such an f as a solution of (2). t

This follows from Friedman [ 9 , Th. 4 and 5 ] provided the second derivatives fy of the f are Holder-continuous. But we can represent the local functions f by Green's formula, using the fundamental solution of the heat equation Au— du/dt = 0 as in § § 9 - 1 0 below. The required Holdercontinuity is then established by standard techniques from the properties of the potentials involved (Pogorzelski [24], Dressel [7], Gevrey [ 1 1 , No. 8 ] ) . 0

(C) Let ft: M^>M' satisfy (2) ; the subscript ( refers to the deformation parameter (we will always indicate explicitly differentiation with respect to t). Then from (4) of § 2 B we have

dt

J"

^ '' dt ^ !

M

SM I

dt

*1.

If D/dt denotes covariant differentiation along paths in M', then for each P € M the curvature veetor of the path i - » f ( P ) is given by D{df /dt)/dt. (

t

PROPOSITION. If ft- M-*M' satisfies (2), then the energy E(f ) is a strictly decreasing function; i.e., dE{f )/dt < 0 except for those values of t for which r(fi) = 0 . Furthermore, its second derivative expresses the average angle between the tension field and the curvature vectors of the deformation paths: t

t

LEMMA. If we let

Let f :M^W t

be an arbitrary deformation for

(£((„,ti).

28

136

J A M E S E E L L S , J R . AND J . H . SAMPSON.

then 1

dt

JM

9

^ftr'Vfl*/

2

P V « /

7

2

This follows from a direct calculation of d e{f )/St and an application of Green's divergence theorem. Because the first integral of the right member is always non-negative, we have on appeal to the above proposition the t

THEOREM.

If / , : M—*W satisfies the heat equation ( 2 ) , then

9

JM

2

dt

^dx* \dt/'

dx> V a * /

O) iff

Hi

2

2

In particular, if M' has non-positive sectional curvature, then d E{f,)/dt 3 0. If t is a value for which equality holds, then r{f,) is a covariant constant; i. e., dx* \

(2)

dt

J

:0 for all P£ M and ( l g i g n ) ,

COROLLARY. / / M' has non-positive sectional curvature and if f, satisfies for all i S i „ , then

at (D) We have seen in § 5 that in supposing M' contained in a larger manifold M" we do not alter the energy of a map / : M -> W. That suggests that we still have control of the energy and the tension of deformations of / which take place in a normal tubular neighborhood N of M' in M". If f: M'—*M" denotes the imbedding, then the induced tangent vector bundle is f- 3 (If) — 3 ( J f ) © *ft (M", W), where the second summand is the normal bundle of M' in M". Then 'ft. —» M' is Riemannian-conneeted in the sense of § 3 A . There ia a canonical vector field : N-^-'U covering the projection map ir: N-*M' defined by assigning to each Q" 6 N the unique vector (Q")eM"(v(Q")) such that exp, (p(Q")) =Q". Suppose now that ft'. M-±N is a smooth deformation ( i ^ i < f i ) . Then p ° / is a vector field on the map *•">/,, and using the harmonic integral l

P

P

tQn

0

t

29 HARMONIC MAPPINGS OP R I E M A N N I A N MANIFOLDS.

137

theory of § 3 applied to the vector bundle 9l —> M', we can define its Laplacean HP°U)-

Let

where D/dt is the covariant derivative in N along the path i r ° / . (

We now establish the following stability property of deformations. T H E O R E M . Let N be a normal tubular neighborhood of M' in M", and suppose that f : M^*N is a smooth deformation ( ( ^ ( < i ) . / / L{p°f ) is always horizontal in ff~ (M") and if /,„: M-*M', then f : M—>M' for all t ^ t < i, t

0

l

t

(

a

t

Proof.

lC

We apply Green's theorem to u, v € A"(M, {v°ft)~ fl)

•

J

*l= — I
The hypotheses imply that °f L(p<> f,)y = 0 for all t, so that t!

o=

f

< °/,,A( °f )>*i— P

P

(

(

&*h^fc*fj$m..

Therefore * 4 f

f< °ftMp°f )>*i

*i=

P

— I.e.,

I

r

t

\d(p °f*)\**i

go.

z

| p o / ( i * l is a non-negative, non-increasing function of t, and it

is zero for t = t„. We conclude that p ° ft = 0 for all t g t < t precisely that every f maps M into M'. 0

lf

which states

t

7.

Global equations.

(A) We now occupy ourselves with the problem of replacing equations (1) and (2), which in terms of local coordinates on M and M' are local systems of equations, by some much more tractable global systems. Remark. Assume for the moment that we have an isometric imbedding w: M'-*Ri for some q, which we can always do by a theorem of Nash [21], Then as in Proposition 5B we find that equation (2) is satisfied for a deformation /(: M —> M' when and only when the composition W = w o f, satisfies the t

condition that the vectors LW, = A V T — a r e perpendicular to M'; see (

30 138

JAMES E E L L S , J H . AND J . H . SAMPSON. q

Lemma 7B below. When expressed in terms of tbe coordinates of R , that condition gives rise to a global parabolic system of equations of tbe type (2). On the other hand, the assumption of an isometric imbedding apparently affords no real simplification in our theory, and we will not make it. We proceed with an elementary imbedding convenient for our purposes. Suppose that M' is smoothly and properly imbedded in some Euclidean space Ri by a map w: M'^*R". LEMMA. Given such an imbedding of M' in K«, it is always possible to construct a smooth Riemannian metric g" = (g"a*)i£a,i£q on a tubular neighborhood N of M' so that N is Riemannian fibred. Proof. Let N be any tubular neighborhood of M' constructed using the Euclidean structure of R"; let IT: N—>M' be the projection map. It suffices to construct an appropriate smooth inner product in each space R (P') for all P' £ M', for we can translate that tangent space to any point Q'€ N along the straight line segment (necessarily contained in N) from P' = ir(Q') to Q'. We take g' in M'(P') and the induced Euclidean metric in the (Euclidean) orthogonal complement of M'(P') in K ( P ' ) , and take their sum in R « ( P ) . q

9

y

x

m

I n terms of local coordinates (y , • • • ,y ) on M' that metric can be described as follows: Write w(P') =• (w (P'),- • - ,vfl(P')); then l

c dw° Qw^

2k dy° ay

s

is the metric on M' induced from the Euclidean metric. Let i" be the unique solution of Z« dy

a

dy

B

3

~

V '

a

dw

fl

We have the duality relation -^^( " = Sa , and the metric tensor is 0

Then so that g" does induce g' on M'. For any vector veR"(P') %v"-— = 0 ( l ^ a S m ) we have ^,v't ^ 0 also, so that a-i oy a=i dw a

a

b

satisfying

31

HARMONIC MAPPINGS OF RIEMANNIAN MANIFOLDS.

139

Tims g" has the desired properties. (B) L E M M A . Let /,: M—*M' be any smooth deformation, and let W,: M-*N be the composition w°f = W . Then T ( / J ) =3/,/0f if and only if r(W )—dW /dt is perpendicular to M'(f (P)) for all P£M. t

t

t

t

t

Proof. The argument is essentially that of Proposition 5 B ; we choose local coordinates on M' and obtain ^ M / . ) * - f ) = ^ < w , ) ' - ^ f

I n terms of the coordinates of Ri the differential of the projection map IT: N—* M' has components *«?OK fdw . Its covariant differential in terms of the metric g" is a

=

^ The map : N-*R«

aSfc ~

r

"

a

o

A

/

(

1

=*

&

c

' =

9

)

•

defined {as in § 6 D ) by p(Q') =Q' — v(Q') assigns to

P

each Q' £ iV a vector perpendicular to M ' ( T T ( C ) ) , and pa" + o " = l r

V>

D

Pas'" + ™a& + r"o6" = 9. For its restriction to M' we obtain

where w ° = dvf/dy". For each a, the right member defines a vector perpendicular to M'; because the w " span the tangent space to M' in which they lie, we obtain the a

a

L E M M A . For any vectors u,v£M'(Q') ire p^Wv" is perpendicular to M'(Q'). LEMMA.

the vector whose components

For any map W,: M^*N with image in M' let £ be the vector

with components e^LWS—Tr^WtfW.W, where L = \ — 9/3f is the heat operator on M. Proof.

As in Proposition 5 B we have L{p{w )y^p/L(w y+^^w^w ^ t

t

l}

Then £ is tangent to M'.

32 140

JAMES B E L L S , J B . AND J . BT. SAMPSON.

(5)

But p(Wf) =0, whence the left member vanishes. the right member is a normal vector, we have

Since the second term in

It follows that

for l^d^g.

Thus J has normal component 0.

PROPOSITION. A map f: M-*M' position W = w°f satisfies

satisfies (1) if and only if the comi

(1)

AW°=* °w >w Y' ab

t

i

in terms of local coordinates on M. A deformation f : M~*M' W, — w°fi satisfies

(t < t < (,) satisfies (2) if and only if

t

a

1

(2)

LiW^^^'WtfWtjY

( i < t < (,). 0

Proo/. It suffices to establish tbe equivalence of (2) and (2). that we take any deformation f and compute

For

t

by the second perpendicular Conversely, if our equation must be 0.

(L(W y—

* <,°W 'W fg

t

a

ti

t

ci

y

>

lemma. I f IT, satisfies (2), then T { W ) ( P ) — dW,(P)/dt is to M'(f (P)), whence by the first lemma f satisfies (2). / , satisfies (2) and we define £ as in the third lemma, t h e n shows that £ ( P ) is perpendicular to ^ ' ( / i ( P ) ) , whence it (

t

t

(C) The following result is an application of Theorem 6D. I n order to have a proof avoiding the use of vector-bundle-valued harmonic forms, we can staTt with (5) and substitute (2) to obtain I (P (W,))

•

mt ( ~ ~ c

+ p T" <) C

AB

WteWtW,

a

using the projection relation p ir„ — it^p^. It follows that " - L( (W,))' so that a

P

P

= 0,

33 HARMONIC MAPPINGS OP R I E M A N N I A N MANIFOLDS.

141

2 -A< <wO)° = ^ 2 ( , { F , ) < r . P

P

Applying Green's identity (as in Theorem 6D) we find that i4f2(p(W )°) *l=-

f

2

(

az

jM

o

2

*l, r

JM

and the conclusion follows: THEOREM. (

0

£ i < ti.

Let W : M->N

be a smooth deformation satisfying (3) for

t

If W 8.

to

maps M into M', then so does every TF { i = t < f

0

ti).

Derivative bounds for the elliptic case.

(A) Under suitable curvature and metric restrictions on M' we shall establish derivative estimates important for the solution of (1) and (3), or equivalently, of (1) and (3). Our starting point is the following result, essentially established in §3C.

LEMMA.

Any solution f, of (3) has energy density e(f ) t

satisfying

d t

(6)

11

-g'aBh-UfR , where fi{ft) is the fundamental form of ft and where R' py$ and R > are the components of the Riemannian curvature tensor on ffl* and of the contravariant Ricci tensor on M, respectively. ( B ) For the elliptic systems (1), (1) we invoke Green's formula for the operator A on M. Let r(P, Q) denote the geodesic distance between points P, Q of M. Since M is compact (without boundary), there is a constant a>0 such that r*(P,Q) is of class C™ for r*{P,Q) < 3a. Let 3a. Set p (P, Q) = <j>(r (P, Q)); this is positive for Py^Q and is of class C. The function F(P,Q)=« (P,Q)-^=^i (P, i&tm~m i

a

2

2

P

(

=

o

g

p

(

3

)

is a parametrix for the operator A, wheTe 1/n is (n—-3) times the surface of the unit (n — l)-sphere. (7)

«(P)= f JM

Green's formula is [u{Q)-A F(P,Q)-P(P,Q)-Au{Q)'\*l , Q

0

34 143

J A M E S E E L L 8 , J E . AND J . H . SAMPSON.

this holding for any function u(P) of class C on M (see Giraud [13], Bidalde Eham [ l ] ) . Consider now a solution of (1), and suppose that the Riemannian curvature of M' is non-positive. Prom (6), in which it is merely necessary to suppress the term de(f )/dt, there follows at once t

Ae(/)

—Re(/),

where R is a constant (independent of the solution in question). Since F(P, Q)^0 (a and

f JM

l*oF(P,Q)+R-F(P,Qne(f)(Q)*U-

By using the osculating Euclidean metric at a point of M one can show that | A F(P, Q) I const. X p(P> Q)'"* - Hence for some constant A, 2

Q

f JM

e(f)(P)^A

F(P,Q)e(f)(Q)*l . Q

Iterating this k— 1 times we obtain f JM

e(f)(P)^A*

F (P,Q)e(f)(Q)*l . k

Q

where the F are denned inductively by F = F and k

±

Fu(P,Q)

=

f F _ (P,Z)-F(Z,Q)*l JM k

l

(t>l).

z

If k > ft/2, then F is bounded (see Giraud [12]), and we have tbe foUowing k

THEOBEM. If M' has non-positive Riemannian curvature, then there is a constant 0 such that e(f) < C-E(f) for any harmonic mapping f: M—>M'. (C) Green's function G{P,Q) for A can be written in the form G{P,Q) =F(P,Q) -\-F'(P,Q), where F' is of class C" for P^Q and has a singularity of order lower than that of F for P^Q (Giraud [12], Bidalde Rham [1 ] ) . G [P, Q) is symmetric and of class C" f or P Q and satisfies A G(P,Q) =& G(P,Q) = V-\ where V is the volume of M. Green's formula (7), with G in place of F, is P

(8)

Q

f

«(P) = F - ' J

u(Q)*l-f

M

G(P,Q)-Au(Q)*l . 0

•/ M

Now let tj be a compact coordinate neighborhood on M, V its interior, and let P e U have coordinates (a; ,• • • ,x"). Write 1

3S HARMONIC MAPPINGS OF RIEMANNIAN MANIFOLDS. i

G (P,Q)=dG(P,Q)/dx ;

G (P, =

i

143

r-G{P,Q)/dxW

U

Using normal coordinates on M one can show that there is a constant 0 such that | \ (9)

G^Q^^C-r(P,Q)-'^

|Gw(P,C)|^C-r(P,g)-» — Gi(Q,Z)\^C-r(P,Q) [r(P,Z)-''* a

I Gi(P,^)

1 a

1 a

+

r(Q,Z)-"' - ]

for P, Q G U, a being an arbitrary but fixed number with 0 < a < 1. If it is a solution of Au = f on M for some function /, then from (8) we have (for P£U) f

MP)

G (P,Z)

f(Z)*l .

i

z

JM Hence IMP)— for P,Qe.U.

f JM

MQ)\^

\G (P,Z)—G (Q,Z)\-\f{Z)\*U i

i

Using the last inequality of (9) we obtain

M wheTe

C = C-sup f JM

+,

Q

1

a

[r(P,?)-" - -f-r(g 2)-"* - ]*l . j

z

Suppose now that / is Holder-continuous with exponent a and Holder modulus M (f). I.e., Jl/ {f) = s u p I f(P) —f(Q)\-r(P,Q)- . It is a classical result a

a

a

M of potential theory that the function f G {P,Z)-[f(P )-f{Z)]*l JM l

0

(P,P 6t7)

z

0

has a derivative with respect to x> at the point P , given by 0

f JM

G (P ,Z)-[f(P )~f(Z}]*l , u

0

0

z

and by (9), this integral is majorized by 0-M (f)-

| JM

a

r(P ,Z)-"^*U. 0

On the other hand, the function = 0, and so f Gi{P,Z)-/{P )*1 -'Af U

K

= 0.

is a constant, since

We conclude that

36 144

JAMES E E L I . S , J E . AND J . H . SAMPSON. i

(11)

\9*u/0t dxf\^C-M (f)

in U.

a

(This is an interior estimate of the type given by Hopf and Schauder (Hopf [ 1 4 ] , Schauder [26], Miranda [18]), but differs from them in that the magnitude of u is not involved.) (D) We apply these results to a harmonic mapping f:M—* the right member of ( 1 ) by F":

M'.

Denote

At this point we impose some boundedness conditions on the embedding of M' in R«, conditions which are automatically fulfilled if M' is compact. Namely we assume that , , | | ^ O, |W/aw" 1 g 0 on Jlf', ' Aids^^ds'^AzdsS, a

0

(

2

where C , At, A denote positive constants and where ds denotes the line element induced on M' by the usual metric in R'. Again let U be the interior of a compact coordinate neighborhood on M , and let P, Q be points of 13. From (12) and an elementary calculation involving the SchwaTZ inequality for quadratic forms and the equality e{f)P = g" W?W Y 0

2

0

,

ab

)

l

\F°(P)—F°(Q)\-r(P,Q)-> ( 1 3 )

a

^ B • [5 + # sup | Wf(P)—W (Q)\

-r(P,Q)

t

where B is a constant and e = e(f) —sup{e(f) ( P ) : P e M).

From (1) and

(10),

[ Wf(P) — Wf(Q) \-r(P,Q)-

a

^ C'sup I F" I, M

and plainly | F* | const. X e, in virtue of (12). of M there follows the estimate (14)

From the compactness

M (P=)gB'(e4-^) t t

for the Holder-modulus of F", B' denoting a constant. we have the

Referring to (11),

T H E O R E M . Suppose that M' satisfies the embedding conditions (12). Let U(x\ • • • ,x ) be the interior of a compact coordinate neighborhood on M. Then there is a constant C such that n

t

\d W/dx*d^\^C(Hf)

+ « ( / ) * ) in U,

(IgaSj),

37 HARMONIC MAPPINGS

OF R I E M A N N I A N

for any harmonic mapping f: M—*M'

Z

MANIFOLDS.

145

where e(f) = sup{e (/) (P) : P e ¥ } .

Remark. We point out that second derivative estimates can be obtained from linear theory in another way. Namely, if we write our equation (1) in the form c

c

e

&W + A >W,>' = 0, where A / J =

i

—ir Wfg i,

b

ab

then we have a linear system with bounded coefficients (in compact coordinate neighborhoods), by Theorem 8B. A Holder-modulus for the W* will then give us a Holder-modulus for the coefficients of the linear system, and we can apply Theorem 1 of Douglis and Nirenberg [6] to deduce second derivative estimates and analogous estimates on all higher derivatives as well. The second derivative estimates we obtain here are somewhat sharper, in certain respects; i. e., they do not involve a priori estimates on the magnitudes of the solutions W. 0

9.

Bounds for the parabolic case.

(A) We now embark upon some analogous computations for the operator £ = A — d/dt. The function !

(15)

2

X ( P , a ( ) = {2V^)-^-'" exp(- (P,g)/40 P

is a parametrix for the operator L (p* as in the preceding paragraph). Put jV,(P, Q, i) =L K(P, P

Q, t) = (\ -dfdt)K{P,

Q, t)

P

and N (P,Q,*)= t

f

f'dr

N^P^J

— T)-N{Z,Q, )*U T

(k>l).

It is well known that there exists a fundamental solution H for the heat operator L on any compact Riemannian manifold M, which can be expressed in the form (16)

H(P,Q,t)=K{P,Q,t)+

f- drf

K(P,Z,t

—

T)-N(Z,Q,T)*lz,

where N(P,Q,t)=i.N*{P,Q,t). (See Milgram-Rosenbloom [ 1 7 ] , Pogorzelski [34]). The function H(P, Q, t) is symmetric in P, Q and is positive. I t is of class C" except for P = t = 0;

10

38 146

J A M E S E E L L S , J H . AND J . H . SAMPSON.

and it satisfies L H(P,

Q, t) = L S{P,Q,t)

P

=0.

Q

Its spectral decomposition

is B(P,Q,t)

= F ^ + £esp(— m

Kt)

where the A„ are the non-zero eigenvalues of A, the

f'dr

u(P,t)

Ji*

(17)

+

f

f JM

— T) LU(Q,T)*1

H(P,Q,t

JM H(P,Q,t

0

— U) u(Q,t0)*lQ

(U
z

this holding for any function u(P, t) on M which IB of clasB C in P and C in ( for t ^ t 0

< tt.

( B ) Suppose now that we have a solution / : M—*M' of (2) defined in 0 < f < i j , and let M' have non-positive Riemannian curvature. According to Lemma 8A we have then Le(f,) 3 — Re(f ), R being a positive constant (independent of the solution in question). Since H > 0, there follows from (17) (

t

(18)

f'dr

e(f,)(P)^R

f

•ft*

+

H(P,Q,t—r)e{f )(Q)*lQ T

JM

6 (f )(P) a

t

where (19)

f

*>(/*)(P)=

ff(P,Q,i-M«(/«.)(e)*lo

"'AT

and 0 < in < ( < i , . Iterating (18) &— 1 times we obtain

(20) f dr

+ e (f,)(P)+iRa

f

ff,(P,g,t

—T)< (/ )(Q)*le, e

T

where the Et are defined by H = E and t

H (P,Q,t) k

Cd, f

= J

a

J

BMP^S—*)£(Z,&4*li

(fc>l).

M

From the integral representation it can be shown (see Pogorzelski [24]) that E(P,Q,t)

<j const. X £ - " r ( F Q ) )

n+2a

(O^(gl).

1

39 HARMONIC MAPPINGS OP RIEMANNIAN MANIFOLDS.

147

where a is an arbitrary but fixed positive number less than 1. Therefore H is bounded for k > « / 2 ( 0 £ < £ 1,ft > 1 ) . Consider now our solution / for ( > 1. (

s

rutting (— 1 for f in (30) 0

we have e ( / , ) ( P ) g const.

('>1T t _ I

(SI)

J

f i'(fr)

K>)*1

m

+ S « p t f o ( / r ) ( P ) - [ l + 2 i ? ' C
f

H„(P,g,r)*l ]. 0

For tbe case at band, e is given from (19) by 0

«.(/.) (J*) Since H(P,Q,

f

//{P.V.ii

^/^KO-io.

1) is bounded, we have 6,(ft) (P) g const.

Recalling that j

« ( f ) (<2)*1 M

0

«•(/») ( £ ) is a decreasing function, we obtain finally from

(?1) e(f,) (P) ^ const, f

C<3)*J

(«>!)•

Auy smaller value can be put in for t —• 1 on the right, for example zero if e(f,)(P) is continuous at 1 = 0. Making that assumption, we now obtain an estimate for the range Q ' £ g i 2 | i . I n (20) we now put t„ — 0, getting (821

e(f,) ( P ) £ const. [ f'.ir o

f *S

-U

where now wo have

JM and where <•„(/,) = sup{<\,(/,) (P) : P € -V} : we define e(f ) similarly. But this function is precisely the solution of £<\, = 0 that reduces at i = 0 to *(/o) {.$)• From general principles it follows that ( ' , ( / i ) ( P ) ^ c ( / i ) for / ^ 0 , and so (?'?) gives at once t

e(f,) ( P ) g const. c(f ) 0

(Ogfgl).

We have then the TUKORKM. Let f,:M-*H' le a family of mappings for 0 5£ f < fi satisfying ("-') for 0 < < < / j and ••>urh that the energy density e(f,)(P) is

40 148

J A M E S E H L L 8 , J E . AND J . H . SAMPSON.

continuous at I

0. Suppose that M' has non-positive Riemannian curvature.

Then e(f )(P)^0-f t

J

and

M

e(f )(Q)*l

j o r t ^ K h

a

a(ii)(P)^o-8up{6(/ )(
for

0

o^;sl,

0 denoting a constant which does not depend on the particular solution ft of (5). Remark. Under certain circumstances much sharper estimates can be obtained. With the hypotheses of the preceding theorem, assume further that the Ricci tensor of M is positive definite at every point. From (6) it is clear that de(fi)/dt^-—Ae(ft) at any maximum point of e(/i) on M, A denoting a positive constant. I t follows easily that e(/,) (P) ^ const, e . A t

1

n

(C) Now let (a; ,' • *,x ) be the coordinates of P in the interior U of a compact coordinate neighborhood 0 on M. And suppose that our solution of (2) and the first-order space derivatives are continuous at fl — 0. Then from (2), (17) we have W '(P) =

W°(P,t)

t

- - C ' d r f H(P,Q,t- )-F>(Q,T)*U Jo JM

(23)

T

=

+

W '(P,t) t

V°(P,t)+W ><(P,t), 0

where m

m

®

f

-

E(p,Q,t)w(Q,Q)*u,

JM the F"{P, t) being the functions on the right of (2). The first integral V"(P, t) has Holder continuous first-order space derivatives (Pogorzelski [23], Theorem 5 ) : | V '(P t) t

1

— V «{P', t') | ^ const, sup \ F°\ - \r{P, P')° -f-1 f _ < * | « » ] t

(

a being an arbitrary positive number less than 1, the points P and P' both in U. I f we continue with the assumption (12), we shall have | F" | ^ const. X «• The integral W °(P, t) can be differentiated under tbe integral sign (for t > 0) and the derivatives tend exponentially to zero, as is quickly seen from the spectral formula for H. Hence, if the hypotheses of Theorem 9U hold, it follows that the functions W (P, t) have first-order space derivatives which 0

C

41 HARMONIC MAPPINGS OF R I E M A N N I A N MANIFOLDS.

149

are Holder-continuous in P, t, uniformly so for t 5; « > 0:

a

gconst. [e(/ ) 4-sup I W ( ( ? , 0 ) | ] • [r(P,P') M

+ | i — i ' WSJ,

0

(The constant depends upon t because of the behavior of W "(P,t) small t.) 0

for

Referring to (13) we see that (

M

\F'(P,t)-F'(Q,t)\-r(P Q)-

)

>

g const. B ( / ) [1 + «(/„)* + fi«P I W°(Q,0) |] 0

forig*>0. Now from (17) it is clear that (

dr f

H(P, Q, t — T)$(T)*1

q

is a

Jo JM function of t alone for arbitrary ip. Hence for the second derivatives F ( P , i ) = d*V (P,t)/dx fc' we can write c

e

l

( i i

f 'dTf H(P,Q,t-T)[F°(P r)--F<(Q,T)-\*l .

F,,(jV)=a^7

a

M

ni

9

From (Pogorzelski [24], Tb. 3) this is V °{P t)= u

t

('dr

f JM

./ f-e

Jo

Jo

ff . (P,g,i- )[P"(P , )-P"(g r)]*l i i

T

0 T

f


where we assume f ^ 2 * > 0 . The integral I is improper but uniformly convergent. Now | H^P, Q, t) | < const. X frfy{P,Q)- -™ (arbitrary 8, 0 < £ < 1 ) . Using (24) and putting P = P„ in I (we assume ( P ^ e P ) , we obtain an absolutely convergent integral if a, 8 are chosen properly, and there results x

n

B

t

\U\<

e

o^t. e(/„) [1 + sup | W°(Q,0) | - f e(/„)*] if C ^ 2*, Af

since then T in J , will be £ f . For I we have | H-i,j{P, Q,t) \ < const, erW for some positive y and for f ^ e > 0. Hence z

• i-e t-t -7('-T) oV

X

E

c

< const, sup | F | < const. e{f ), 0

using Theorem 9B.

42 150

JAMES E E L L S , J R . AND J . H . SAMPSON.

T H E O R E M . Suppose that M' satisfies the imbedding conditions (13). Let ft satisfy the conditions of Theorem 9B, and let (x ,- • • ,x ) be the coordinates of a point P in the interior of a compact coordinate neighborhood U on M. Given t > 0, there is a constant C, independent of the solution f of (2), such that 1

n

t

c

d*W (P,t) dx^x' for i^t,

0)\:Q£M}

< C e(f ) [ 1 + gtf )» + BupfJ W'(Q, 0

0

where e(f ) — sup{e(f ) (P) : P € M}. 0

0

Semarh.

Since I H{P,Q,t)*lQ is a constant, the functions W"(Q,0) JM appearing in the foregoing estimates can be altered by arbitrary additive constants without affecting the validity of the estimates. For example, one could replace W"(Q, 0) by the function minus its average value, say W , with the result that the term sup | W (Q,0)\ in Theorem 9C would be replaced by sup| TP"(Q,0) — W«\. c

a

10.

Successive approximations.

(A) Let W(P,t) and W'(P,t) be two solutions of (2) in 0^t
0

W ( P , ( ) — W">(P,t)

f ' dr f H (P, Q, t - r) CNft r) - tf* (Q, r) ] *1 , 0

Jo

•*M

c

where F" F' are the respective right members of (2). }

Z(t)

= s u p | W°(P,t)—

Set

W'°(P,t)\

M,c e

+ [sup(W —W '°) (Wf— t

{

W'iBtni.

M,B c

c

U(t) =e(f )

+e(r.)

From the constitution of F and F' it is easily verified (by an argument similar to that foT (13)), account taken of (12), that \F»—F"°\
+ *(/,)» + a

Hf>)y

For 0 £ ( g l we can write E(P,Q,t) >+>"' and | dR{P, Q, t)/dx* \ < const. ir«r{P, Q)--™° (0 < a < 1 ) ,

43

HARMONIC MAPPINGS OP R I E M A N N I A N MANIFOLDS.

151

where in the latter the constant depends upon the particular choice of local coordinates x . of course. Now let A denote an upper bound for the quantity U(t) in some fixed time interval 0 g t f = i . From the integral expression above there follows easily, for \ < a < 1, i

2

X(t)

< const. X A J ' {t —

and we conclude that X(t)

r)-"X{T)d

T!

vanishes for small t.

Hence the

T H E O R E M . Let f, and ft be two solutions of (3), both continuous along with their first-order space derivatives at ( = 0. If fo = fo, then the two solutions coincide for all (relevant) t > 0. COROLLARY.

Any solution of ( 2 ) enjoys the semi-group property along

the trajectory of each point P £ M.

That is, if we write ft[P) = T, (f ) , then

T (U)=T (f )=Tt(T (f )). UT

t

T

T

0

( B ) For tbe solution of ( 2 ) we now turn to the system of non-linear integro-differential equations ( 2 3 ) associated with ( 2 ) . 1

Let / : M - » M' be a mapping of class C , given by global mapping functions T P = ( W V •

W).

For v ^ O define W'=

W°-(P,f)= f Jit

(25)

{W--\- • • ,W'^)

by

H(P,Q,t)W°(Q)*l

Q

and C

W" (P, t) (26)

=

— C dr f HiPtQ^—^-F'-^iQ^^U Jo JM

+

W^iP^),

where b

F-"{P,t)

ii

=ir °(W)-W r-«W - g , ab

t

)

the functions TTab" as in ( 2 ) . Set y = sup [ 2 WV^TF/'V']*v

From ( 2 6 ) Wf<(P,t) (27)

f'

Jo

dT

C n (P,Q,i JM t

—

r)-F*-^(Q,r)*lQ+W^(P,t),

44 152

J A M E S B E L L S , J E . AND J . H . SAMPSON.

where the subscript i denotes differentiation with respect to a system of local coordinates x* at P. We recall the estimate I

a

Q,t)\<

At~ r{P, Q)-"-^<*

( 0 ^ i ^ l ; i < « < l ) ,

A denoting a constant which depends in general on the local coordinate system. Now let B denote an upper bound for the quantities | w b | in some compact neighborhood V of the image f(M) in the tubular neighborhood N C R«. I f W- €. U' for 0 g t% t, then from (27) we conclude that 0

a

1

where C is a constant which does not depend upon the given mapping f. Put y — sup y r

O^i^t,

n

Then y < Kt-oy^ v

a

1 0

If K^- g^

1

S £ and Ke - ^

+ y*,

K = BG/(1 — *).

% £, then

From a transparent induction it follows that, for sufficiently small positive *, we shall have W(P,

a

() 6 V and y„ ^ ^K^~ for 0 g * g t, y = 0,1,2, • •

In particular, the defining equation (26) makes sense for all v, provided Now put e

X „ ( i ) — sup I W- (P, f)

W - ^ ^ P , f) I 1

fl

• f [sup(W^= —W^" '") (Wy- — W y - ^ f l * ' ] ! . From the definition above, j^.e—j'F-i.n._,

W o f t 0

( r p » - i ) ( 1 F ^ » 1 P ^ I > — Wf-i*Wf-%.b)

gti

+ [».»"(W) — ^ " ( T F ' - ^ l W ^ W j - V ' . We can suppose that the constant B occurring above is also an upper bound for the quantities | d-n „ /dw | in the neighborhood IT. The preceding formula c

a

d

45

HARMONIC MAPPINGS OF BIEMANNLAN MANIFOLDS.

153

then gives us | p > - J W * | < C'BZ (y P

P

+

+

C a constant independent of the given mapping /. Hence, | F^ — F'- -"1 < G'BXill/K*-* 1

2

+ 1/4SV- "]

From (36), (27) and the estimates cited for H(P,Q,t) obtain f Z„(r)

X^{t)
=C"Z (t). v

and ff (P,Q,tO we (

a

((-T)- dr

(0g^ ), t

Thus if

X„(i) ^ s u p Z ^ r )

(O^rgi),

then

and so the series

0

converges for all sufficiently small (. This shows that our successive approximations W* and their first-order space derivatives converge uniformly on M (for small t). Hence, the F"- also converge. Set W" = lim W' and F" = lim¥"•" ( c — » » ) . Thus W'(P,t) has continuous first-order space derivatives Wi' (for sufficiently small t, of course), and W / —> WV (?—»a>), so that C

C

,e

J*>(P,^^(rfjTf^y). From (26) there follows at once (28)

W(P,t)

f'dr

f

3(P,Q,t

Jo

— T)-F°(Q,T)*l

Q

+

W°-°[P,t).

JM We conclude that the functions W have Holder-continuous first order space derivatives—uniformly Holder-continuous for any small elosed (-interval not containing zero. Therefore the functions F " ( P , t) are Holder-continuous with respect to the space variable, and so tbe functions W° of (28) satisfy equation (2) for all positive t in the interval in which the successive approximations converge (see Pogorzelski [24]). The ff" are moreover visibly cone

46 154

JAMES EEIXB, JE. AND J . EI. SAMPBON.

tinuous with their first order space derivatives at (— 0, and W"(P, 0) = W(P). From Theorem 7C we recall that the point W(P,t) — (W ,- • • ,W*) must lie on the manifold M' for all t in the interval of convergence. 1

THEOREM. Let M" be a compact subset of M'. Then, for any continuously differentiate mapping f:M^M' such that f(M) lies in M", there is a positive constant i , depending only on M" and the energy density e(f) such that (2) has a solution f, for O g ( g d which is continuous at ( = 0, along with its first order space derivatives, and which coincides with f at t = d. It only remains for us to inquire into the length of the (-interval for which the successive approximations converge. First of all, the upper bound B for | ir ," | and j (hatf/dm* | figuring in the foregoing proof can be taken once for all to be valid in some neighborhood U" of M". Therefore the constant K can be fixed, and a as well, of course. The < must satisfy Kf ~ y £ J , and to see what this entails we must look briefly into y„. Consider then a solution u of the heat equation on M, as a map u: M—*R. We suppose that u and its first order space derivatives are continuous at ( = 0. The argument of Theorem 9B holds in this situation. If we put p^g uiu, , then p(P, t) ^ Csupfp (Q, 0) : Q € M], where C is a constant depending only on M. Applying this to the functions W , we conclude that there is a constant C\ such that y„ < C\ • e(f). Thus < depends only on M" and the magnitude of e(f), and the same is true of the quantity C" involved in the estimates of the X . The assertion of the theorem then follows at once from those estimates. In § 2B we described the harmonic character of C maps in terms of their tension fields. For C maps we have the ai

l

a

a

il

j

0,c

v

1

1

COEOLLAEY. Let f: M—*M' be a continuously differentiate mapping for which the energy is a minimum with respect to small variations. Then f is harmonic. For let ft be the corresponding solution of (3) guaranteed by the preceding theorem. The energy E (/,) is continuous at ( = 0 and by assumption it must be non-decreasing for small (. But E{f,) is always non-increasing, so that dW/dt^O for small t. Thus (2) reduces to (1). (C) THEOEEM. Suppose that M' has non-positive Riemannian curvature, and that it satisfies the embedding restrictions (12). Then for any continuously differenliuble mapping f: M—*M' there is a unique solution f,

47 harmonic

mappings of

riemannian manifolds.

155

of (2), defined for all t^Q, which is continuous along with its first order space derivatives at t — 0 and which coincides with f at t = 0. Such a solution exists for small t, by Theorem 10B, and it is unique, by Theorem 10A. Let t, be the largest number such that a solution of the required sort exists for 0 f§ t < t„ and supposes that t± is finite. From Theorem 9B it is clear that the right members of (28) cannot become unbounded for O g i < ( i and consequently the images fi{M) (0^(<(,) all lie in a compact subset of M'; we recall that M' is always assumed to be complete. On the other hand, Theorem 9B shows that the energy density e(f,) remains bounded, and therefore by Theorems 10A and 10B there is a fixed positive number such that any f, can be continued as a solution of (2) into the interval ((,14-<,). This contradicts the definition of (D) If W is not compact, then solutions of (2) may very well become unbounded as t —*oo, as in the Example. Let M' be tbe manifold obtained by revolving the graph of a positive strictly decreasing smooth function v = v(u) around the u-axis; let <> / denote the revolution angle. For a map / : S —» M' of a circle S parametrized by the central angle 0 our heat equation is 1

H 3t dt

=


v'v" 1 + (v')

2 +

M-

2

+

/3uV \d&)

1

vv' 1+{»')*

(HY W

v 86 di) '

If / satisfies initial conditions du/d8 = 0, = 8 when t =- 0, then so does the solution / , for any subsequent time. If we take v{u) = 1 4 - e~ , then R'i2i2 = — ( « " + l ) / ( e + 1) < 0, and the heat equation reduces to u

I u

du e" 4- 1 St ~ e 4-1 * Ju

Thus e" 4- u — 2 logic" 4- 1) = ( 4 - const.; in particular, u—>oo as t—>w. We note in passing that there are no non-trivial closed geodesies on W, so that there are no harmonic representatives in any non-trivial homotopy classes of maps S —» AT. The following result shows that solutions must remain bounded if M' satisfies certain conditions at infinity. 1

48 156

J A M E S E E L L S , J E . AND J . H . SAMPSON.

THEOREM. Let M' be as in the preceding theorem and suppose further that j w | •TTab'tw) —» 0 uniformly as \ w j —>oo, where \ w | = sup \w°\. Then 0 every solution of (2) is bounded. Set W(t) = sup| W"(P,t)\,

W(t) = inf I W°(P, t)\,

c

U(t) =

2

f 2 (W ) *l.

In virtue of Theorem 9B, the difference W(t) — W(i) is bounded. Hence if our solution is unbounded as (-»<», then that is true of all three of the quantities above. Supposing that to be the case, let us denote by A& the set of all ( for which U(t) > fc. The Aj are then all non-empty and each As must contain at least one t = t at which dU/dt^.0. Now from (2) we have k

2 W°AW — 2 o

o

-n at

c

1

2 W Tr ° WfWjY ab

Hence by Green's Theorem idU/dt=—

f (gradW) *l — 2 f WV^W^W.VM. •J M aJ M 2

For large values of f* we have a plain contradiction, since the right-hand side must be negative. 11.

Harmonic mappings.

(A) We can now apply some of the results established above to prove the existence of harmonic mappings, even though we do not know whether the solutions of the parabolic system (2) converge in general as t—»oo. THEOEEM. Let M' have non-positive Riemannian curvature and let f :M—*M' be a bounded solution of (2), 0 < i < o > . Then there is a sequence d, U, t , • • • of t-values such that the mappings f, converge uniformly, along with their first order space derivatives, to a harmonic mapping f. t

a

k

From Theorems 9B and 9C it is clear that the mappings / , and their first order space derivatives form equicontinuous families. Hence there exists a sequence t„ t - • • such that the mappings f — f, converge uniformly, with their first order space derivatives, to a continuously differentiable mapping f. From (2) and (8) we can represent the f by the formula S!

k

k

k

49 HARMONIC MAPPINGS OF RIEMANNIAN MANIFOLDS.

157

c

where as usual F stands for the right member of ( 2 ) . Now fix c and temporarily put

X

dW-

The dW{P, t )/dt are bounded as &-»oo, by Theorems 9B and 9C, and so the u and their first derivatives are bounded. Hence the u form an equicontinuous family, and we can suppose that the sequence t t ,- • • is chosen so that the u converge uniformly, say to u. Now let (?„ denote the i>-th iterate of the Green's function G. We have k

k

k

l7

2

k

f G,(P,e)«(G)*l=lim f J M

G»{P,Q)u (Q)*\ k

h J M It J M

0,1

If v 4 - l > f i / S , then G is bounded. But the dW'(P, t )/dt the mean to zero as h—*oo, by Corollary 6C. Thus W 1

converge in

k

r

G»(P,Q)u(Q)*l

= 0,

J M

and so u = 0 because of the positive-definite character of G. Therefore, passing to the limit in the equation above, we get for the limit mapping f the formula

where

T P « ( P ) = 7 - » j" W'(Q*i— f G(P,Q) JM JM e

F (Q)—\imF»( Q, k L

h)

F<(Q)*1 , 0

=- '(W)WfWW. rM

C

From this it follows (as in § 1 0 B ) that W (P) has Holder-continuous first derivatives, and therefore f" is Holder-continuous. Consequently the W(P) satisfy (1). COROLLARY. Let M ' have non-positive Riemannian curvature and let f: M M ' he a continuously differentiate mapping. Let f, be the solution of (2) which reduces to f at ( = 0. / / / , is bounded as t—>oo, then f is homotopic to a harmonic mapping f for which E(f') ^E(f). In particular, if M ' is compact or satisfies the conditions of Theorem 10D, then every continuous mapping M—>M' is homotopic to a harmonic mapping.

50 158

JAMES EELLS, JR. AND J . H. SAMPSON. COROLLARY. If M' is compact and has non-positive Riemannian curvature,

then every homotopy class of mappings M—* M' contains a harmonic mapping whose energy is an absolute minimum. For in any homotopy class we can choose a minimizing sequence of harmonic mappings f ft, etc., by the preceding corollary. From Theorems 8B and 8D it follows that we can select a subsequence (same notation) which converges uniformly along with first derivatives to a continuously differentiable mapping /. Then E(f) = lim.E(ft), and f is harmonic by Corollary 10B. u

(B) THEOREM. Let M have non-negative Ricci curvature and let M' have non-positive Riemannian curvature. Suppose that M' is compact or that it satisfies the conditions of Theorem 10D. Then any continuous map f: M^*M' is homotopic to a totally geodesic map. Furthermore, 1) if there is at least one point of M at which its Ricci curvature is positive, then every continuous map from M to M' is null-homotopic; 2) if the Riemannian curvature of M' is everywhere negative, then every continuous map from M to M' is either null-homotopic or is homotopic to a map of M onto a closed geodesic of M'. This is a combination of Theorem 3C and Corollary l l A . COROLLARY. Let M be a compact smooth manifold admitting a Riemann structure g with non-positive Riemannian curvature. Then M does not admit any Riemannian structure g with non-negative Ricci curvature unless that curvature vanishes everywhere. 1

Proof. It suffices (by passing to the two leaved orientable cover if necessary) to consider the case that M is orientable. I f there were such a metric g, then the identity man (M, g) —> (M, g') would be homotopic to a totally geodesic map. That map has degree one, and therefore M cannot have any point of positive Ricci curvatuie relative to g. Remark. A special case of Theorem 11B (Part 1) can be obtained without the use of harmonic theory. Namely, assume 1) that M has positive Ricci curvature everywhere; then by a theorem of S. Myers [30], the fundamental group ^ ( j y . w i ) is finite;

51

HARMONIC

MAPPINGS

OF RIEMANNIAN

MANIFOLDS.

159

2 ) that M' is anr complete Riemannian manifold with non-positive Riemannian curvature. Then the homotopy groups m!) = 0 for t ^ l , and •w,(M',m'} has no elements of finite order. It is well known that the homotopy classes of maps of any arcwise connected space 11 into are in natural 1-1 correspondence with the conjugacy classes of homomorphisms 7r,{Jf, m) —>ir,(il', m'). But clearly in the situation at hand every such homomorphism is trivial, whence every continuous map M —* Zl' is null homotopic. COLUMBIA UNIVERSITY, T H E JOHNS HOPKINS UNIVERSITY, T H E INSTITUTE FOR ADVANCED STTDY.

REFERENCES.

[I] [2]

P. Bidal and G. de Rham, ' ' L e a formes differentielles harmonique," Commentarii Mathematici Helvetia, vol. 19 (1946-71, pp. 1-49. S. Bochner, ''Harmonic surfaces in Riemannian metric," Transactions of the

[3]

American Mathematical Society, vol. 47 (1940 J , pp. 146-154, , "Analytic mappings of compact Riemann spaces onto Euclidean space,"

[4]

Duke Mathematical Journal, vol. 3 11937), 339-354. , " C u r v a t u r e and Betti numbers in real and complex vector hundles," Ren-

diconti del Seminario Mateniatico di Torino, vol. 15 (1955-0), pp. 225-2-53. 15] C . Chevalley, Lie Groups I. Princeton, 1946. [6] A . Douglis and L . Xirenberg. "Interior estimates for elliptic systems of partial differential equations," Communications on Pure and Applied Mathematics, vol. 8 (1955), pp. 503-538. [71 P. Dressel, " T h e fundamental solution of the parabolic equation I , " Duke matical Journal, vol. 7 (1940 ) , pp. 186-203. [8] L . P . Eisenhart, Riemannian Geometry. Princeton, 1926.

Mathe-

[9] A. Friedman, " I n t e r i o r estimates for parabolic systems of partial differential equations," Journal of Mathematics and Mechanics, vol, 7 (1958), pp. 393-418. [10] F . B . F u l l e r , "Harmonic mappings," Proceedings of the Xational Academy of Sciences, U.S.A., vol. 40 (19541, pp. 987-991. [11] M. Gevrey, " S u r les equations aux derivees partielles du type parabolique," Journal de Mathematiques et Appliquees, b. 6, vol. 9 (1913), pp. 305-475, G. Giraud, " S u r le probleme de Dirichlet generalise," Annates de I'Ecole Normale Huperieure, vol. 46 (1929), pp. 131-245. [13] R . Hermann, "A sufficient condition that a map of Riemannian manifolds be a [12]

fibre bundle," Proceedings (1960), pp. 236-242.

of the American

Mathematical

Society,

vol. 11

52 160

JAMES EELLS,

J R . AND J . H . SAMPSON.

[14] E . Hopf, " TJber lien funktionalen, insbeaondere den analytischen Character der Losungen elliptiacher Different! algleichungen zweiter Ordnung," Mathematische Zeitschrift, vol. 34 (1931), pp. 194-233. [15] A . Liehnerowicz, " Quelquea theoremes de geometrie differentielle globale," Commentarii Mathematici Belvetid, vol. 22 (1949), pp. 271-301. [16] C . Maxwell, A Treatise on Electricity and Magnetism, Oxford, 1892. [17] A . Milgram and P. Rosenbloom, "Harmonic forma and heat conduction I , " Proceedings of the National Academy of Sciences, U.S.A., vol. 37 (1951), pp. 180-184. [18] G. Miranda, Equazioni alle Derivate Paraiali di Tipo Ellittiao, Berlin, 1955. [19] C . B . Morrey, " T h e prohlem of Plateau on a Riemannian manifold," Annals of Mathematics, vol. 49 (1948), pp. 807-851. [20] S. Myera, " Riemannian manifolds with positive mean curvature," Duke matical Journal, vol. 8 (1941), pp, 401-404.

Mathe-

[21] J . Nash, " T h e imbedding problem for Riemannian manifolds," Annals of Mathematics, vol. 63 (1956), pp. 20-64. [22] I . Petrovsky, " O n the analytic nature of solutions of systems of differential equations," M atematiieskiy Sbornik, vol. 47 (1939), pp. 3-70. [23] W. Pogorzelaki, " Propriatea des integrates da l'equation parabolique normale," Annates Polonici Mathematici, vol. 4 (1957), pp. 61-92. [24]

, " E t u d e de la solution fondamentale de l i q u a t i o n parabolique," Ricerche di Matematica, vol. 5 (1956), pp, 25-56. [25] S. Sasaki, " O n the differential geometry of tangent bundles of Riemannian manifolds," Tohoku Mathematical Journal, vol. 10 (1958), pp. 338-354. [26] J . Schauder, " Numerische Abschitzungen in elliptischen linearen Differentialgleichungen," Studia Mathematica, vol. 5 (1934), pp. 34-42. [27] D . Spencer, Mimeographed notes, Princeton University.

A n n . Inst. Fourier, Grenoble 14, 1 (1964), 61-70.

ENERQIE ET DEFORMATIONS EN QEOMETRIE DIFFERENTIELLE (') par James E E L L S , Jr. et J . H . SAMPSON (Columbia University et Strasbourg.)

1. Introduction.

De meme qu'une fonction reelle definie sur une variete entraine une decomposition (cf. theorie de Morse) de son domaine (sous certaines hypotheses, bien entendu), une fonctionelle definie sur un espace d'appiications peut donner lieu a une decomposition agreable de celui-ci moyennant certains systemes differentiels associes a la fonctionnelle. Les methodes principales sont la theorie des varietes differentiables de dimension infinie et la theorie des semi-groupes dans les espaces hilbertiens. Ici nous voulons donner une breve description de ces methodes telles qu'elles se presentent dans 1'etude de certains problemes geometriques du calcul des variations sur les varietes riemanniennes. Pour notre formulation nous avons profile de plusieurs conversations avec MM. S. Smale et A. H. Taub. 2. Un probleme general.

Dans ce paragraphe nous voulons esquisser une formulation du calcul des variations adaptee a une classe tres etendue de problemes qui font intervenir des derives d'ordre superieur d'inconnues dont le domaine est une variete difTerentiable (') Ce travail a fite finance partiellement par la National Science Foundation (U.S.A.) et l ' U . S . A r m y Research Office (Durham).

54 62

JAMES E E L L S ,

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donnee. Parmi les problemes que nous envisageons certains sont obtenus par un procede d'iteration, done forcement des problemes d'ordre superieur. Le langage des espaces fibres est le plus naturel pour notre formulation. Soit X une variete riemannienne compacte. Pour ecarter des complications nous la supposerons orientee, sans bord, et de classe C". Soit ^ : V —>• X un fibre difTerentiable sur X ayant comme fibre une variete difTerentiable, et notons J (%) —*• \ lefibredesft-jetsde sections de \ (on ecrira \ tantot pour V, tantot pour 1'espace fibre tout entier). Par densite wariationnelle d'ordre k sur £ nous entendons une fonction difTerentiable / : J*(ij) -> R. Une section suffisamment difTerentiable f du fibre S; determine d'une facon evidente une section f' de J*^); et en appliquant a celle-ci la densite f on obtient une Tonction reelle f(f') sur la variete X. L'integrale de cette fonction, etendue sur X moyennant sa structure riemannienne, sera notee Iy(tp). Done L peut etre considere comme fonctionelle sur une classe donnee V de sections suffisamment differentiables de !;. Le choix de T joue un role important. Avec un choix convenable et sous des hypotheses raisonnables sur la densite / (convexite, par exemple), la fonctionnelle L determinera par ses varietes-niveau une decomposition de son domaine V, decomposition cependant beaucoup plus delicate dans le cas des espaces fonctionnels que dans le cas des varietes ordinaires. II n'est guere besoin d'insister sur sa relation etroite avec la theorie de M. Marston Morse. Les fonctionnelles L que 3'on vient de deerire ont une grande importance pour plusieurs problemes de l'analyse globale et de la geometrie differentielle. Nous en verrons quelques exemples plus loin. Precisons maintenant les espaces fonctionnels que nous voulons considerer. Pour chaque entier ft(0 ^ k < oo) notons C*(!j) 1'espace des sections de classe C* du fibre !;, et notons H(!;) 1'espace des sections de !j dont les derivees d'ordre <J r (par rapport a un systeme de cartes sur X) sont de carre sommable. On munit ces espaces des topologies evidentes. Si r > n/2 + ft, n designant la dimension de X, chaque element de H(c;) est une section de classe C , et on demontre que l'application canonique H (^) C (Z) est continue (lemme de Sobolev; voir [10, Lemme 3.2] ou bien [5]). k

r

r

r

k

55 ENERG1E

E T DEFORMATIONS EN GEOMETRIE D I F F E R E N T I E L L E

63

On sait le resultat suivant depuis quelque temps; voir [3] pour les idees essentielles : k

T H E O R E M E . —• Pour 0 ^ ft < GO Vespace C (c]) variete" differentiable, localement un espace de Banach, sion infinie en general. Pour n/2 < r -< GO Vespace une variete differentiable et les espaces de fecteurs admettent une structure hilbertienne.

est une de dimenH {^) est tangents r

r

Comme choix de T, les espaces H (£) semblent presenter certains avantages sur les C*(ij). En particulier, nous utilisons I'existence sur H (^) d'une structure riemannienne alin de construire le champ v d'Euler-Lagrange pour la fonctionnelle L, ce qui nous met a meme d'etudier les lignes de plus rapide decroissance de L. Or notre probleme general est bien l'etude de ces trajectoires de Vj, c'est-a-dire des applications r

s

$ : R X H'($) - * H'(S)

satisfaisant a l'equation

0

y£$).

II sera surtout question de leur convergence vers des points critiques de L lorsque le parametre t tend vers l'infini. Mais il convient de noter que meme I'existence et Punicite des trajectoires voulues presentent de grandes difficultes en general. La construction du champ Vj peut se faire aussi pour des valeurs de r inferieures a n/2, bien que H (£) ne soit probablement pas une variete alors. Par exemple, pour r — 0 et J; un fibre riemannien, le champ v est une application de H°(^) dans H°(TJj), ou T!; est lefibredes vecteurs tangents a f; et cette application est caracterisee par la formule r

f

(iM,

«> = ( I / U » =

oh fj> e H°(!j), oil u est un element de differentielle formelle de L.

v.h

H°(T^),

et oii (lf)

r

est la

3. Quelques excmplcs de deformations.

A. D E N S I T E S QUADRATIQUES. — Si \ i V -> X est un fibre fibre vectorielle, il en est de meme de 1'espace J ($), des ft-jets, et H {^) est un espace vectoriel admettant une struck

a

r

56

64

JAMES E E L L S ,

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ture hilbertierme. Dans ce cas on peut considerer les densit£s variationnelles f: J*(i;) R qui sont quadratiques dans chaque fibre. Le champ Vj d'Euler-Lagrange associe a une telle densite definit a chaque point 0 e H{S;) un operateur lineaire, discontinu en general. r

]

1. — Prenons pour 5 le fibre T'* (X) formes differentielles exterieures sur X, et posons Exemple

f(

X les

+ W j ,

9

oil d* est la codifferentielle pour la structure riemannienne donnee sur X. Alors I/: H (^) -> R n'est autre que l'integrale de Dirichlet pour les formes differentielles. Munissant H ^) du produit scalaire de H°(£), on obtient vfy) = A
1

etudiee par Milgram-Rosenbloom [7] fournit un semi-groupe a un parametre de deformations (R>0) X IT(5)-> H'tf), a partir d'une forme 9 donnee. Mais si H ($) est muni du produit scalaire deduit de l'integrale de Dirichlet, le champ vt est, a chaque point de H (^), un operateur borne"; et l'equation p

r

ot

a

;

definit un groupe a un parametre de deformations d'une forme f donnee. Cette theorie fut utilisee par Morrey-Eells [10] dans leur demonstration du theoreme de Hodge. Exemple 2. — Chaque operateur lineaire D : C°(5) C°°(rj) d'ordre k defini pour desfibresriemanniens S; et r) sur X determine une densite variationelle f-a d'ordre k,

57 ENERGIE

E T DEFORMATIONS E N GEOMETRIE

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65

et le champ d'Euler-Lagrange est v (f) = D*D T'Y. D'autre part,
r

T

T

TY

et

T'

1

On pose alors f = ° T'f o i oT
Exemple 1. — Prenons pour o* la fonction \/a , oil a„ est la n-ieme fonction symetrique elementaire. Les points critiques de la fonctionnelle L qui en resulte sont les soi-disant « sous-varietes minimales », bien que celles-ci ne soient ni varietes ni minimales en general. Desormais nous nous bornerons au cas
S

o u

oil g (resp. k) est la metrique sur X (resp. sur Y). L'integrale de faif) considere comme fonction sur X sera notee simplement E(• R. C'est la fonctionnelle d'energie; elle est tres importante en geometrie differentielle. Ses points critiques sont les applications X —> Y dites harmoniques. Ces applications sont preColloque Grenoble.

5

58 66

JAMES

EELLS,

J R . E T J . H . SAMPSON

cisement les zeros de l'operateur non lineaire elliptique du deuxieme ordre K^ =

A^ +

n (Y)gg;V

(1
E

oil les a? (resp. les 9") sont des coordonnees locales sur X (resp. sur Y ) , ou A est l'operateur de Laplace-Beltrami sur X , et ou les rja(Y) sont les symboles de Christoftel de Y , m etant la dimension de Y . Pour H ( X , Y ) muni de la structure metrique de H°, les 1

solutions de l'operateur non lineaire de la ckaleur (i)

sont les analogues des lignes de plus rapide decroissance.

Nous avons utilise la solution fondamentale de l'equation de la chaleur sur X pour demontrer [12] que l'equation (1) a une solution unique pour un certain intervalle de temps se reduisant a une application donnee de classe C pour t = 0. Certaines majorations des derivees d'une solution eventuelle de (1), obtenues par une methode geometrique due a Bochner dans un autre contexte, nous permirent d'etablir le 1

THEOREME. — Soient X et Y des varietes riemanniennes fermees, et supposons que la courbure riemannienne de Y soit non positive. Alors il existe dans chaque classe d'homotopie d'applications $ : X - > Y une representante harmonique qui est un point de minimum absolu pour I'energie E . Exemple 2. — Dans le cas d'une variete X de dimension 1 (l'intervalle ferme ou le cercle), on sait que I'energie E : H ^ X , Y)

-> R

est une fonction differentiable qui determine une bonne decomposition de 1'espace H ^ X , Y ) , dont on obtient les estimations de Morse des nombres de Betti de 1'espace des lacets de Y . On en trouve un expose elegant dans Palais [11]. Le theoreme enonce ci-dessus peut etre etabli dans le cadre de la theorie de la fonction differentiable E sur la variete

59 ENERGIE

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EN GEOMETRIE

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67

r

H (X, Y), pour r > ra/2 (n etant toujours la dimension de X). Mais il convient de signaler qu'a defaut de la condition requise sur la courbure de Y, la fonction E peut ne pas donner une bonne theorie de Morse. Par exemple, si nous prenons

la sphere euclidienne (n 3), alors E n'a aucun point de valeur minimum sur les composantes de H (S , S ) des applications de degre ^= 0. En effet, on peut trouver une famille a un parametre T , : S„ «-* S„ d'applications C°° et de degre un telle que lim E(*F ) = 0 (t -*• oo). Mais bien entendu il n'existe pas d'application de degre un dont I'energie s'annule. r

n

n

t

C. E N E R G I E S D'ORDRE SUPERIEUR. — S. Smale et R. Palais ont etudie le principle de Dirichlet dans 1'espace euclidien a n dimensions; S. Smale a pu demontrer I'existence d'un extremum pour une classe de fonctionnelles quadratiques du /r"°" ordre, avec k > n/2. Ce resultat souligne l'importance d'une etude de ce que nous appelerons e^nergie du /c" * ordre E . Done si T*X = T(T* X) est 1'espace fibre here de vecteurs tangents, chaque application 9 : X Y de classe C* induit saft""differentielle T"f : T*X->T"Y, et Ton peut s'en servir, comme nous l'avons fait dans le cas k = 1, pour definir une energie E^'f^). Le laplacien itere apparait comme partie principale du champ d'Euler-Lagrange associe a E \ II y a lieu de croire que des methodes analogues a celles qui nous servirent dans la demonstration du theoreme cite au n° 3 B permettront une generalisation de ce theoreme pour les E " , sans hypothese sur la courbure de Y si k est suffisamment grand. Et l'on peut esperer trouver une theorie de Morse pour E<*>: H*(X, Y) ^ R (A > n/2). m

w

-1

( t

0

D. Q U E L Q U E S REMARQUES. — La notion de o"-energie peut etre appliquee a d'autres situations. Par exemple, si p(g) = (R^g)) est le tenseur de Ricci de la metrique g sur X, alors <Ji(p[g)) = g^Rij = R, la courbure scalaire. Son integrale est la courbure totale de X, consideree comme fonction de g; notons-la T(g). Hilbert a montre [4] que ses extrema satisfont a l'equation R f — ^ R / 2 = 0.

60

68

JAMES

EELLS,

J R . E T J . H . SAMPSON

Voir aussi Taub [3]1, ou est decrite une methode par laquelle on peut normaliser la fonctionnelle T(g) afin d'obtenir pour ses extrema les metriques dites d'Einstein : — g"R/n = 0. Nous ne connaissons pas de conditions raisonnables sur X qui garantissent I'existence d'un extremum de T. Pourtant la nature des estimations figurant dans la demonstration du theoreme du n° 3 suggere I'existence d'une deformation de g dans une metrique d'Einstein sur X si g est a courbure riemannienne strictement positive. (Cf. Berger [1]). Pour la validite de cette assertion il est clair que certaines restrictions topologiques sont necessaires. Par exemple, S X S* et (S X S ) # (S X S ) (somme connexe) n'admettent pas de metrique d'Einstein. Remarquons, pour terminer, que la conjecture de Poincare dit que T (normalise convenablement) a un extremum si X est simplement connexe et de dimension 3. Pour d'autres problemes relatifs a Ia deformation de metriques, voir Calabi [2] et Yamabe [14]. 1

1

3

1

3

BIBLIOGRAPHIC [1] M. B E R G E R , Les spheres parmi les varietes d'Einstein, C.R. 254 (1962), 1564-1566, [2] E . C A L A B I , The space of Kahler metrics, Proc. Inter. Congress. Math., 1954. [3] J . E E L L S , On the geometry of function spaces, Symp. Inter, de Top., At . (1958), 303-308, [4] D. H I L B E R T , Die Grundlagen der Physik (Erste Mitteilung), Nachr. Ges. WLss. Gott., (1915), 395-407. [5] J . L . L I O N S , Equations differentielles operationne'les et problemes aux Hmites, Springer, 1961. [6] P . L A X and J . A . M I L G R A M . Parabolic equations and semigroup, Annals s

of Math. Studies, 33.

[7] A. M I L G R A M and P . C . R O S E N B L O O M , Heat conduction on Riemannian manifolds I, Proc. N.A.S., 37 (1951). [8] C . B. M O R R E Y , The problem of Plateau on a Riemannian manifold, Anncds of Math. 49 (1948), 801-851. [9] C . B. M O R R E Y , Multiple integral problems in the calculus of variations and related topics, Ann. Scuola Nor. Sup. Pisa, (1960), 1-61. [10] C . B. M O R R E Y et J . E E L L S , A variational method in the theory of harmonic integrals I, AnnaU of Math; 63 (1956), 91-128. [11] R. S. P A L A I S , Lectures on Morse theory; Mimeo. Notes. Brandeis Univ., 1963.

61

ENERGIE [12] [13]

[14]

E T DEFORMATIONS

E N GEOMETHIE

DIFF ERENTIELL E

69

et J . H. S A M P S O N , Harmonic mappings of Riemannian manifolds, Am. Journ. of Math, (a paraitre). A. H. T A U D , Conversation laws and variational principles in general relativity, Notes of a lecture at the Summer Syntp. in Diff. Geometry, Santa Barbara, (1962). H. Y A M A B E , On a deformation of Riemannian structures on compact manifolds, Osaka Math. J., 12 (1960), 21-37. J. EELLS

62

lie printed

from

Proceedings of the United Stalea Japan Seminar in Differentia! Ceomelry Kyoto, Japan, 1965

Variational Theory in Fibre Bundles J a m e s E E L L S , J r . a n d J o s e p h H . SAMPSON

11

T h i s i s a b r i e f d e s c r i p t i o n of c e r t a i n m e t h o d s — s o m e

old a n d

s o m e n e w — o f e s t a b l i s h i n g e x i s t e n c e i n t h e c a l c u l u s of v a r i a t i o n s , c o n f i n e d ( f o r t h e p r e s e n t p u r p o s e s ) to a s e t t i n g w h i c h s e e m s propriate

for

R i e m a n n i a n geometry.

r e s u l t s h a v e a p p e a r e d in

[4, 5, 6 ] ;

Some

our

ap-

of o u r m e t h o d s

viewpoint

overlaps

and to

a

c e r t a i n e x t e n t w i t h t h a t ( d e v e l o p e d e s s e n t i a l l y s i m u l t a n e o u s l y ) of P a l a i s a n d S m a l e [17, 18, 23],

1.

Abstract variational theory. (A)

C l a s s i c a l l y , t h e p r i m a r y m e t h o d i n t h e c a l c u l u s of v a r i a -

t i o n s is b a s e d o n t h e t h e o r e m of W e i e r s t r a s s t h a t i f F:

V-*R

is a

l o w e r s e m i - c o n t i n u o u s f u n c t i o n o n a c o m p a c t s p a c e V, t h e n t h e r e i s a p o i n t of V a t

which F

assumes

its

minimum

value.

This

w a s t h e b a s i s of t h e m e t h o d u s e d for e x a m p l e i n H i l b e r t ' s p r o o f of the

existence

of

geodesic

segments,

in

the

potential

theoretic

s o l u t i o n of t h e P l a t e a u p r o b l e m ( e v e n i n i t s e x t e n s i o n b y M o r r e y [12] to R i e m a n n i a n m a n i f o l d s ) , a n d i n t h e a b s t r a c t a r e a t h e o r y of F e d e r e r - F l e m i n g [7) a n d of R e i f e n b e r g [19]. A s a n illustration in the differentiable context, let V be a real Hilbert space

a n d F: V-*R

for s i m p l i c i t y of derivative

of

F

statement). a t xe

a differentiable If

l

d F{x;u,v)

f u n c t i o n (of c l a s s denotes

V i n t h e d i r e c t i o n s u, v,

the

z

C

second

then we have

the

PROPOSITION. / / d*F(x] v, v)^0 for all x,veV, then F is lower semi-continuous in the weak topology of V. Furthermore, the subsets of V which are weakly compact are precisely those which are bounded and weakly closed. T h i s s i m p l e r e s u l t is a n a b s t r a c t i o n of t h e b a s i c w o r k of T o n e l l i 1)

Research partially supported by N.S.F. Contract G P 4217 and G P 1825. 22

63

VARIATIONAL THEORY IN F I B R E BUNDLES

23

and Morrey (see [13]) on convexity properties of a variational density to insure existence in the calcuJus of variations. EXAMPLE. The norm is weakly lower semi-continuous in a Hilbert space V, and any closed convex subset K of V is weakly closed. Thus there is a point of K at minimum distance from the origin in V. This is the essence of Weyl's method of orthogonal projection. (B) That method was greatly broadened by M. Morse [15] using homology theory. The sets F (—oo,b)=Fi for b eR (which are closed when and only when F is lower semi-continuous) define a filtration of the space V, and analysis of the homology groups H(Fi, F ) for a^Lb relative to some fixed coefficient field leads to an abstract definition of critical points of type k^O. In this general form we have a method of obtaining extremals in the calculus of variations, used to a certain extent by Morse-Tompkins [16] and M. Schiffman [21] in their work on the Plateau problem. It is essential that we take into account critical points of type k > 0; for there are standard variational problems admitting extremals which are critical points of higher type, but not admitting any extremal which is a minimum: See (4, Section 4E of Chapter % _1

a

Indeed, from this general topological viewpoint the abstract calculus of variations becomes a theory of the decomposition of a space V by means of a real function, using the order of the real numbers. In particular, we would like to join each point x£ V to a critical point of F by a path (sort of a gradient line) lying in V. There are various types of decomposition that one might consider: 1) filtration of the space V itself by F; 2) filtration of the homology of V by the H{Ft, Ft); 3) decomposition of a suitable cellular space having the homotopy type of V; 4) handlebody decomposition of V (homeomorphically, or diffeomorphicaily, when that makes sense). We will call decompositions of types 2), 3), 4) Morse-Smale decompositions. (C) Our aim now is to present another method, less general, but well adapted to variational problems of differential geometry. In broad terms, we formulate certain variational problems in the context of infinite dimensional differentiable manifolds, and then use ideas of non-linear functional analysis and infinite dimensional

64 24

J. EELLS,

Jr. A N D J . H . SAMPSON

Riemannian geometry to produce solutions of the given problem. The effect will be to construct a decomposition of the appropriate function space by lines of steepest descent; for further motivation in a linear setting we refer to [6]. 2.

General form of the problem.

(A) Let X b e a compact oriented Riemannian M-manifold. We will suppose throughout that X has no boundary, which will have the effect of restricting our variational problems to closed problems; that will simplify our exposition. On the other hand, the modification of our formulation and methods to include boundary value problems, while presenting interesting analytic and geometric features, does not offer severe difficulties. Let r: V^X be a smooth ( = differentiable of class C°°) fibre bundle over X, whose fibre model is a manifold, again without boundary. We let / ( r ) : J*(V)-*X denote the bundle of r-jets of sections of r. A variational density of y of order r is a smooth function / : f'(V)-* R. If C"(V) denotes the space of smooth sections of the bundle J and j':C (V)-*C (J (V)) the r " jet prolongation, then we define the function F:C"(V)^>R through the diagram T

a

a,

C(V)

r

R

f

I'' , C<"U'(V))—>-C°"(XxR) Here the right hand arrow denotes the integration over X of a real valued function with respect to the volume «-form *1 of X: r

F(>P)= [fU 9>l

•

Our first problem is the following: Given a component of C"(V), find in it an extremal of F\ for the nonce, let " extremal" be taken in a homological sense. We arrive at once at a fundamental technical idea of variational theory: Although we have a specific problem formulated in a differentiable setting and we require differentiable solutions, it is most convenient to complete the problem by suitably enlarging the function space C"( V) and extending the

VARIATIONAL THEORY

IN FIBRE BUNDLES

25

function F to the enlarged space. In general there can be many essentially different completions of a problem. We will confine ourselves here to what we will call Riemannian problems: We will use essentially the Riemannian structure of X to construct spaces of sections of r which are often Hilbert space manifolds; our variational problems will satisfy a form of Legendre condition, so that their Euler-Lagrange equations will be of elliptic type. (B) Let us suppose for the moment that r is a Riemannian vector bundle f: W-*X; that means that £ is a smooth vector bundle over X having a linear connection P' and an inner product on each fibre, expressed as a 2-covariant tensor field g$, related by pe.g*=0. For any real number 1 < I J ? < C O let LAW) be the completion of the vector space C°°(W) in the norm Mvif>=Q

,

P

l?U)l '*iJ' .

For any integer r g 0 let LA W) be the Banach space of all sections such that all iterated covariant differentials Fty are of class LAO ^ t £ r). In particular, for p = 2 we have the inner product

giving a Hilbert space structure to LAW). The following results [24] are fundamental, where the inclusion is taken to mean that the natural map is a continuous injection; SOBOLEV THEOREMS 1. LA. W)^ LAW) and

\lq >

for 0^ s
— n/p,

. 2.

LA W ) c CK W) for r > n/p+k .

In the present paper we will need this second result only for k = 0. p = 2. The following application is known [3,5]: THEOREM. Let r: V-*J£ be a closed fibre subbundle of £, and LA V) = {

«/2,

25

J. E E L L S , Jr. AND J. H. SAMPSON

then LAV) is a closed smooth submanifold of the Hilbert space LAW). In particular, LAV) has a smooth Riemannian structure. There are three main points in the demonstration: 1) Use of the inverse function theorem in Banach spaces; 2) use of the first Sobolev theorem to show that the composition of an LA W)-tunction and a smooth function is an Lr'fTTJ-f unction; 3) use of the second Soboiev theorem to insure that the topology in LAW) is larger than the uniform topology. (C) DEFINITION. A variational density / o f order r is (quadratically) Coextensive if the function F admits an extension of class C to a function LAV)-*B. For the case r = ? good conditions to insure that / b e Coextensive have been given by Smale [23]. In the present work we will not worry further about C*-extensivity of a variational density, for in most of our applications F; Lr\V)—*R arises as the restriction of a continuous quadratic form on the Hilbert space LAW), and is therefore automatically C°°-extensive. k

( D ) Given such a variational density / we consider the gradient field t of F on the Riemannian manifold LAV). Its trajectories (fpt)

are the solutions of

= — ?(
or^

relative to the initial Con-

dition p, = ip

a

when

/ = 0.

As in the finite dimensional case one proves the PROPOSITION. If r > M/2, / is Coextensive and / SO, then these trajectories are uniquely defined for all teR. A limit point of (fi) as t—*oa is a differentiable critical point p« of F, and hence represents an extremum in the generalized sense that it belongs tO Lr\V). We are thus in position to formulate the basic Condition (CT): Every trajectory of F has a critical point in its closure. This condition is not always satisfied. It is insured, however, if the sets F* are all compact, or if each trajectory (¥>()iao is relatively compact.

VARIATIONAL THEORY IN FIBRE

BUNDLES

27

If Condition (CT) is satisfied, then we are in position to provide generalized solutions to the problem posed in Section 2A: Starting with any p e L r ( V ) there is a critical point ip^eLr"(V) which, using the Riemannian geometry of L (V) and Sobolev's second theorem for r > M/2, is shown to be arcwise connected to fa- Now critical points tpSLr\V) can sometimes be shown to belong to C™(V), in which case our solution is satisfactory; however, that question of higher differentiability of critical points is a most difficult and only partially developed aspect of the field (see Morrey [13] for a discussion of recent results). We will content ourselves by remarking that in our examples in Section 3 below, all critical points p « e L r ( F ) are actually in C~{V). !

0

2

R

!

(E) Our first problem is only the first step in a much richer theory, of the sort outlined in Section IB, Palais and Smale have taken this stronger viewpoint by requiring Condition (C). Given any sequence (pjJtju in L \V) such that F{ai converges T

in

Lr\V).

Condition (C) implies Condition (CT) and can be viewed as motivated by it; for if {co, and inf {|r(
t

(

REMARK. Smale [23] has used Condition (C) in case r is a vector bundle to establish the existence of L -solutions of a certain non-linear Dirichlet problem. We cite also the recent work of F . Browder [1], applying (C) in the context of Lusternik-Schnirelmann 2

r

28

J . E E L L S , J r . AND J . H . SAMPSON

category. (F) To us the most unsatisfactory aspect of our theory is the requirement that r > «/2. Presumably, for 0 g r ^ M/2 the function space L \V) is not a manifold in general. That seems to mean, roughly speaking, that it is easier to solve variational problems whose integrals involve high derivatives in a prominent role than it is to solve low order problems. (This was recognized also by Palais [17, §16] and Smale [23, Condition (3)].) T

3.

Examples of the general theory.

(A) EXAMPLE 1. Suppose £: V-*X and n: W-*X are vector bundles over X and A an r** order linear elliptic operator from the sections of £ to sections of v. Then the variational density /(p) = l/2|.4 'l , relative to some inner product in q, is quadratically C=°-extensive, and defines a continuous quadratic form F: LAV)-* K( £ 0). Condition (CT) is satisfied, and in fact (as a consequence of the linearity of the problem) the trajectories converge to points of the kernel of A, a finite dimensional subspace of C™(V)
r

As an application, we cite the following well known result; Given iireC™(W), there is a
(B) We obtain a large variety of variational problems by taking the situation of Example 1 and restricting / to a closed subbundle r of f having compact fibres (this last condition is not essential; it suffices to have fibres having some metric boundedness conditions). For expositional reasons we will discuss only the following special case.

69

V A R I A T I O N A L T H E O R Y IN F I B R E B U N D L E S

29

EXAMPLE 2. Take for $ the trivial bundle XxR" and for r the product manifold X x F c X x J ? ' ; we will view the sections of r as maps from X to Y. We set A — (d+d*) and define the r energy E (
th

1

T

&(¥<) = 1/2 j \(d+d*)' \**\ V

. s

Its restriction to the closed submanifold Lr (Xx Y) is smooth (when r > n/2). Its extremals are called the polyharmonic maps of degree r {or r-harmonic maps) from X to Y. They are known to be smooth. THEOREM. / / Y is compact and r > n/2, then every map from X to Y is homotopic to an r-harmonic map. If r = \ or if Riem (Y)^.0, then the same conclusion holds. This was proved a few years ago [4, 5, 6]; a problem similar to the first assertion but involving boundary values was solved by Palais [17], and his approach can be applied to produce that assertion without essential difficulty. Furthermore, it is not difficult to generalize both assertions to non-trival bundles r. EXAMPLE. If n = l, the 1-harmonic maps are just the closed geodesies of Y. For this case our proof in [4] does not use any restriction on the sectional curvature of Y. A treatment of closed geodesies—in particular their Morse theory—as critical points of Ei has been given by J. McAlpin [10]; the starting point is verification of Condition (C), modifying the presentation of Palais [17]. It follows immediately (as McAlpin has shown) that there is a nontrivial closed geodesic on Y, thus producing a simple proof of the theorem of Fet. This approach has subsequently been taken up by Klingenberg [9], whose paper claims many important new results. If Y is not compact, then Condition (C) is never satisfied; however, if YIs complete and satisfies certain natural conditions at infinity, then Condition (CT) is satisfied. For r > 1 the r-harmonic maps are the poly geodesies, which on Y play the role of polynomials, just as geodesies play the role of straight lines.

70

30

J . E E L L S , Jr. AND J. H . SAMPSON

EXAMPLE. I f n = 2 and r=l, the energy function E\ is called the Dirkhlet-Douglas integral. The 1-harmonic maps are called minimal surfaces of Y (although of course they are neither minimal nor surfaces, in general). We feel that this case is very special in our theory. For instance, E, is a conformal invariant of X; as another instance, LftXx Y)czLAX x F ) for all 1 1. If we could establish the existence of critical points of E for all r > 1, perhaps we could then conclude the existence of a critical point for Ei itself. (Compare the method in Example 3 below.) T

(C) Still another example of a variational problem obtained by restriction of a quadratic form has been given by Yamabe [25]: EXAMPLE 3. Take r as the trivial bundle XxR F: Li\XxR)->R by the quadratic form ( « S 3 ) Fiv) =

and define

4(w - 1 ) \d
where R:X->R is the scalar curvature of the Riemannian metric of X. Now for every number l^q<2n/(.n — 2) we have LiHXx R) f l is defined by f,{RYamabe has shown that smooth extrema >p, exist for all 1 £a ?<2«/(« — 2), and that as q increases in that 0

71

VARIATIONAL THEORY IN FIBRE BUNDLES

31

range the

4.

Variation theory and cobordism:

A problem.

(A) Thus far in our presentation we have emphasized homotopy aspects of our problem: Proving the existence of extrema in a given component of a function space of sections. That is a somewhat narrow view, and not at all in the proper spirit of many classical variational problems. (For instance, the Plateau problem starts with an immersed 1-manifold B in a given Riemannian manifold Y, and then asks for a minimal surface in Y spanning B. There are no additional restrictions on the surface, such as requiring that it should have a prescribed topological character.) It seems to us that the following problem—here loosely formulated for maps of closed manifolds, but capable of generalization—is of importance. (B) DEFINITION. Suppose the Riemannian manifold Y fixed, together with a positive integer n. Let us consider pairs {
th

PROBLEM. Fix r £ l . Under what conditions is there an extremal pair in a given cobordism class ? 1

For instance, let us take « = 1; then in the picture {

•See note on p. 33a,

72 32

J. E E L L S , Jr. AND J. H. SAMPSON l

and {

A cobordism of (

V

Bibliography [1] Browder, F.: Infinite dimensional manifotds and nonlinear elliptic eigenvalue problems, Mimeo. notes, Univ. of Chicago. 12 ] Dieudonne, J.: Foundations of Modern Analysis, Academic Press, 1960. [ 3 J Eells, J.: On the geometry of function spaces, Sym. Inter, de Top. Alg., 1958, 303-308.

[ 4 ] Eells, J. and J. H. Sampson: Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86 (1964), 109-160. [5] : Energie et deformations en geometrie differentielle, Annales de lTnstitut Fourier, 14 (1964), 61-70.

[6]

: Analyse fonctionnelle non lineaire et geometrie differentielle, Sem. Ehresmann.

73

VARIATIONAL THEORY IN F I B R E BUNDLES

[7]

F e d e r e r , H . and W . F l e m i n g :

33

Normal and integral currents, A n n . of

M a t h . , 7 2 (1960), 458-520. [8]

Gaffney:

T h e heat equation method of Milgram and Rosenbloom for

open Riemannian manifolds, A n n . of Math., 6 0 (1954), 458-466. [9]

Klingenberg, W . :

[10]

McAlpin, J . : bia

(11]

U n i v . , T h e s i s , 1965.

M i l g r a m , A . and P . Rosenbloom: I,

[12]

Closed geodesies, Mimeo. Notes, 1 X 5 .

Infinite dimensional manifolds and Morse theory, ColumHarmonic forms and heat conduction

P . N . A . S . , 3 7 (1951), 180-184.

Morrey, C . :

T h e problem of Plateau on a Riemannian manifold, A n n .

of Math., 4 9 (1948), 801-851. [13j

•

;

Multiple integrals

in the calculus of

variations,

Colloq.

L e c t u r e s A . M . S . 1964. [14]

and J . E e l l s :

A variational method in the theory of harmonic

integrals I , A n n . of M a t h . , 6 3 (1956), 91-128. [15]

Morse, M . :

Functional topology

Gauthier-Villars, [16]

and T o m p k i n s : critical type,

117]

Palais, R . :

and

abstract

variational

theory,

minimal surfaces of

general

Paris, 1939. Existence of

A n n . of M a t h . , 4 0 (1939). Morse theory on Hilbert manifolds,

Topology, 2

(1963),

299-340. [18]

—

and S. Smale:

A generalized Morse theory,

Bull.

A.M.S.

165-172. [19]

Reifenberg, E . : Solution of surfaces.

the Plateau problem for m-dimensional

A c t a M a t h . , 1 0 4 (1960), 1-92.

[20]

Rothe, E . H . :

[21]

Schiffman, M . :

Gradient mappings.

[22]

Shibata, K . :

B u l l . A . M . S . , 5 9 (1953), 5-19.

T h e Plateau problem for non-relative minima, A n n ,

of Math., 4 0 (1939), 834-854. On the existence of a harmonic mapping, Osaka Math.

J . , 1 5 (1963), 173-211. [23]

Smale, S . : problem,

[24]

Morse theory and a non-linear generalization of the Dirichlet A n n . of Math., S O (1964), 382-396.

Sobolev, S . :

Applications of functional analysis, T r a n s l . Math Mono-

graphs, 7 (1964). [25]

Yamabe, H . :

On a deformation of Riemannian structures on compact

manifolds, Osaka, Math. J . , 1 2 (I960), 21-37.

Cornell U n i v e r s i t y Johns Hopkins University

74 33a

*Note to p. 31

Yamabe's assertion is correct, although his proof is not. A coherent exposition of the subject, describing carefully the contributions of many mathematicians, is given by Lee.J.M. and Parker, T. H.: The Yamabe problem, Bull. Amer. Math. Soc, 17 (1987), 37-91.

75

ToptAosj

Vol.

IS. pp. ZBMH. Pefgmwii

Pi™.

1974.

RESTRICTIONS

Printed In GreiL Brilun

O N HARMONIC MAPS

O F SURFACES

I . EELLS and J . C . WOOD (Received lOSeptember 1975) Dedicated 10 Professors S.-S. Chern and H Wtiilney. i l . STATEMENT OF THE MAIN RESULTS

LET X and V be closed orientable smoolh surfaces, and
w„+(2o ic)wM w

=0,

(1)

being the Euler-Lagrange equation associated to the energy functional

The purpose of this notet is to prove the THEOREM. If ip:X-*Y is a harmonic map relative to Riemannian metrics g and h, and if e(X) + \d\ e(Y)\>0, then ipis ± holomorphic relative to the complex structures determined by g and h. l

Here e(X) = 2-2p and e( Y) = 2 -2q denote Euler characteristics; and d„ is the degree of is ± holomorphic. In particular, if the Riemann surface X has no meromorphic function of degree p, then there is no harmonic map (S, h) with ^ 2, whatever the metrics g and h; and are essentially the only ones with those degrees. In contrast, Smith 110] has constructed examples of harmonic maps of degree 0 from a flat torus to the Euclidean sphere which are surjective; and others whose images are proper subsets with interior. Finally, Corollary 2 implies that there is no harmonic map tp: (T,g)-+{S,h) of degree ±1, whatever the metrics g, h. (Earlier it had been shown by Lemaire [8] and Uhlenbeck that there was no such map giving an absolute minimum of the energy). Example. Let X be a hyperelliptic Riemann surface of odd genus p s 3. Then there is no meromorphic function ip:X-*S of degree p, since every such

(S,

h) of degree p, whatever the metrics (compatible with the complex

structures).

By way of contrast, A. M. Macbeath has shown us that meromorphic functions of degree p do exist on all other Riemann surfaces X of genus p 3=2. tA special (and Ihe moat important) case of (hat result was anounced a[ [he Summer Course in Complex Analysis, Trieste 1975{ 14]. I' is a pleasure Irj record our rhanks loM. J. Field. L. Lemaire, and A. M. Macbeath for their comments during the preparation of this note.

76

264

I. EELLS AND J. C. WOOD

Application. We give an analytical proof of trie following topological theorem of H . Kneser[7]: If q»2, then for any continuous map ip?.X-*Y, we have \d \e(Y)s: e(X). Namely, we introduce an arbitrary metric g on X, and a metric k of negative curvature on Y. and appeal to the existence theorem[3] to obtain a harmonic map a>: (X,g)-*(Y, ft) which is homotopic to thus d. = aV Now if \a\\e( Y) < e ( X ) , then e(X) + |d.e( Y)\ > 0, so that

|

[ W f f f - ' n l l s p - l . T h u s | d , ( q - l ) | = |p*H'(n)| = | H V - n | ' S p - l ] . Remark Lemaire notes that with our present theorems[81—we can respond to the question:

knowledge—especially

his

existence

Giuen closed orientable surfaces X, Y and a homotopy class of maps ip:X-*Y, can we find metrics g, h on X, Y relative to which there is a harmonic map if homotopic to ^i? The answer is yes in all cases except whenp = 1, q = O.and = 1. In that case the answer is DO. SI. PROOF OF THE THEOREM Given a smooth map y : X -* Y, its differential dip: T(X) -»T( Y) extends to a complex linear map d'
T{X)® C;

IX]. Corresponding to the decompositions T'(X) = V"{X)®

we have

3uoip: T'^X^T'^Y),

3e.,
H

T°'(X)

see [6, Ch.

V(X)-*T°-'(Y).

These can be interpreted as sections of the complex line bundles ncfXlOcff-T^r)

and

TUX)®
respectively. We note that 3i.o^=0iffp is anti-holomorphic, c V , f = 0 i f f ip is holomorphic. The following result is standard; PROPOSITION 1. If the sections i'l.w- and 3 ip have only finitely many zeros, then 0lI

Index 0,. a>) = -e(X)

+

o

d,elY)\

Index lo .,
(2)

a

Here Index(i) = the sum of the zeros of the section s (having only finitely many zeros) of the complex line bundle f over X, counted at each point x according to the local degree of s at x. Then Index(s) = c(f), the (first) Chem class of f; see [11, Part III]. For the first assertion of the Proposition, we take H . o ( X > ® c » > " T " { Y ) for f Now [A, Chapter I ] l

c«.®f ) = c«,)+c« ), 5

)

c(f*) = - c ( f ) ,

|

e(p- ^)=d.e(f).

Since we have canonical isomorphisms T t . ( X ) = T * ( X ) and T ' ^ O O ^ T
a

s

complex line

Index (a,. >) = c{TUX)® -'T(Y)) = clTU(X)+c{
cV

r

PROPOSOION 2. If q>:X->Y is harmonic, then the (Ifir-pan of the tensor field ip*h is a holomorphic quadratic differential on X; we denote it by i).. Furthermore, n . = 0 iff ip is ± holomorphic.

77 RESTRICTIONS ON HARMONIC MAPS OF SURFACES

265

Here

T,, = A ( z ) d z ' = c r V ( z ) ) | r f f -

0)

The partial differentials Sijof and 3*.i

(4)

Y is harmonic but not ± holomorphic, then neither 3,.nip nor Bo.iip has a

zero of infinite order. m

For otherwise the holomorphic function a(z) = o{\z\ )

as z-*0

for all m »0; but that would

imply that a = 0 , so that by Proposition 2, ip would be ± holomorphic. LEMMA. If ip: X - » Y is harmonic but not ± holomorphic, then in isothermal charts 3x$

3wiBz = Az

+o(\z\") for some m 3 0 and complex number A ^ 0 ;

Bwl3z = Bz" +o{\z\") for somen 1*0 and BkO-

(5)

h

Proof. Taylor's expansion to m' order for the first gives 3wldz-Q (z,z) m

where Q

+ R(z,g),

(6) m

m

is a homogeneous polynomial of degree m ==0, and R is 0(1^| ) as z - * 0 .

The preceding Corollary shows that Q „ # 0 for some m. Substituting (6) in ( 1 ) gives

31 Now 3Q l3z m

a

3z

is either a homogeneous polynomial of degree m - 1 or is identically 0. Each of the 1

other terms is o(|z|"'~ ). It follows that BQaldz = 0. Therefore Q„ is a holomorphic homogeneous polynomial of degree m, and consequently has the form Az"

The case of Bo.iip is handled

similarly. Remark. A local expansion theorem for harmonic maps, their differentials, and Jacobians was given by Wood [12, T h 1.4.8; 13], based on work of Hart man-Win tner and Heinz. Our needs here are more modest, and the above direct argument suffices. PROPOSITION 3, If ip:X-*Y

is harmonic and not ± holomorphic, then the zeros of 3,. ip and a

de.tip are isolated and of strictly positive index. Proof. Taking an isothermal chart centered at a zero x of Si&p, we obtain from (5) Bw/dz = Az" +o(|z|"') with m >0; thus d,.r,ip has a zero of index m at x. Similarly for the zeros of do. 1 (p. Proof of the theorem. Assume that

0, r

one of the right members of (2) is negative,

contradicting Proposition 3. REFERENCES 1. H. BEHNKE and F. SOMMEB: Theorie der ano/jiiicfitn Funkiionen finer kompiexen Vaiandtrikhen. Springer-Verlag, Berlin (1955). 2. S. S. CHFRN and S. I. GOLDBERG: On the volume-decreasing properly of a class of real harmonic mappings. Am. 3. Morn. 97 (1975), 133-147.

3. J. EELLS and J. H. SAMPSON: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86 (1964), 109-160. 4. F. HIRZEBRUCH: Topological Methods in Algebraic Geometry. Third EdiLion. Springer (1966), 5. M. GBPSTGNIIADEB and H. E. RAUCH: On extremal quasi-conformal mappings—1, Proc. natn. Acad. Sci- 40

(1954).

808-812.

and K. NOMIZU: Foundations of differential geometry—II. lnterscience (1969). 7 H. KneSEit: Die klcinste Bedeckungzah] imnwrhalb einer (Classe von Flachenabbitdungen. Moth. Ann 103 (1930). 347-358. 8. L . LEUATRE: Applications harmonique! de surfaces. C.S., Paris. 280 (1975), Ser. A. 897-899. 9. J . MtLttOR: On the existence of a connection with curvature zero, Comm. Math Httv. 32 (1958). 215—223¬ 10. R. T. SMITH: Harmonic mappings of spheres. Thesis, Warwick duly 1972), 6. S KOBAYASW

2fiS

I. EELLS AND I. C WOOD

11. N. SmwtOD: Topology oj Fibre Bundles. Princeton (]9J1). 12. J . C. WOOD: Harmonic mappings between surfaces. Thesis, Warwick (June 1974). 13. J. C. WOOD: Singularities of harmonic mappings, lo appear. 14. J, C. WOOD: Harmonic maps and complex analysis, Proc. Summer Course in Complex Analysis. Trieste (1975). 15. H-H Wu: The equidistribution theory of holomorphic curves. Ann. Math. Stud (1970). University of Warwick. Brighton Polytechnic.

79

The Surfaces of Delaunay James Eells

1. Background

2. Ron I elk's of a Conic

In 1841 the astronomer/mathematician C Delaunay isolated a certain class of surfaces in Euclidean space, representations of which he described explicitly Ml in an appendix lo that paper M. Strum characterized Delaunay's surfaces vanationally; indeed, as the solutions to an iso peri metric problem in the calculus of variations. That in turn revealed how those surfaces make their appearance in gas dynamics; soap bubbles and stems of plants provide simple examples. See Chapter V of the marvellous book |81 by D'Arcy Thompson for an essay on the occurrence and properties of such surfaces in nature.

The first step is to derive the equations describing the trace of a focus F of a nondegenerate conic f as ^ rolls along a straight line in a plane (Perhaps these derivations were better known a century agol) We examine various cases separately f IS A PARABOLA:

More than 130 years later E. Calabi pointed out to me thai the solutions to a certain pendulum problem of R. T, Smith \7] could be interpreted via the Gauss maps of Delaunay's surfaces [2}. And Eells and Lemaire 141 found that the Gauss map of one of those surfaces produces a solution lo an existence problem in algebraic/differential topology. The purpose of this article is lo retrace those steps in an expository manner—as a revised version of [2)

James Eells

MATHEMATICAL INTtl.LICENCtfl VOL 1. NO I *> I9R7 Spiin^Tr-V-rl^ UVK/ York 5 3

80 Here A is the vertex of f The line PK is tangent to ( at the point K. The following properties are elementary: (1) Correspondingly marked angles are equal; (2) FP is orthogonal lo PK. Thus we obtain FA = FP cos Z AFP = FP cos C PFK. Now we change our viewpoint and Ihink of the tangent line PK as the axis—the i-axis—along which the parabola t rolls. We denote the ordinate of F by y; and observe that cos I . PFK =

m

as

where a denotes the angle made by the tangent with the x-axis. Thus setting c = FA, we obtain the differentia] equation

V

ds

Thai equation describes the shape of a flexible inextensible free-hanging cable—thereby explaining its name. In that context we can obtain the equation of the catenary as the Euler-Lagrange equation of the potential energy integral 1

2

V +• y' dx.

subject to variations holding fixed the length integral PI £Vl +

dx = L.

Indeed, from general principles we are asked to find A real number a and an extremal of the integral

l(y)

- Q,

c - y

(2.1)

- e cosh xle.

~

describes the rate of change of abscissa of F with respect to arc length s; i.e., •m_

V = 2 (<*" +

=

j'W

1

+

v"

+

"yVl + y'

:

dx

lis Euler-Lagrange equation has first integral y -

/(I + ay)- . /

r;

, for (J

i

R.

The equation of the catenary is derived from this, choosing suitable normalizations. The curvature of ( is measured by the amount of turning of its tangent That is expressed by the Gauss map of f into the unit circle, given by x —• where dx c COS o. = — = - , ds y

V l + y"

-

Its solution is the catenary

A*

54 THE MATHEMATICAL IATGITGENCEB VOL. *. NO. 1.11?

81 The Gauss map of the roulette of the parabola is injective onto an open semicircle

The locus (of either focus) will be called the unduiary:

f !S AN ELLIPSE:

Its Gauss map is given by x —* a . where y

1

- —

f/V

y + (-> COS a - + " It maps i onto a closed arc of the unit circle. There are two limiting cases, which are perhaps best handled separately: When b -• a the unduiary degenerates to a straight line, the locus of the centre of a circle rolling on a line. And where b —• 0 the unduiary becomes a semicircle centred on the .r-axis.

1 O F'

j

t IS AN HYPERBOLA:

Here F and F are Ihe foci of t j the 0 is its centre. The line PKP' is tangent to i at K. Letting n and IF denote the lengths of the semi-axes of f , we obtain the following properties: (1) TM + F K = 2a > 0; (2) the pedal equation PF • FT 6]);

y\\i' O

;

= b (see [°. Ch.VHI

/

(3) the normal to the locus of F passes through K P^-

Again using PK as i-axis. S= - sin i. FKP = cos L FTP - ^~

In analogy with the Case of the ellipse, we have 4=-su\i-

(1) FK - P K = 2o > 0;

F'KP- = cos L FTP' = ~-.

2

(2) Pf - P T = b . Thus we obtain the following differential equation for the locus of F, given as a firsi integral of an EulerLagrange equation:

From these we derive •

i

to

z

u ± lay

2

y y' = b , so Ihat

f

- 2«y| +

By analyzing all cases and taking a & b. we obtain 1

2

2

b = 0.

(2.3)

The loci of the two foci fit together to form the curve which we shall call the nodary:

V = 0.

V t 2 « y ^ + b = 0.

§j

ai)

The solutions to that differential equation can be given explicitly in terms of elliptic functions; see [J], [5, pp. 416-416].

82 lis Gauss map x -* a, is governed by !

y "& cos a, = ~ Thus f - y'fy = ± e , where b is another real paramlay eter. Consequently, The Gauss map has no extreme points, and direct verification shows that it is surjective. A roulette of a conic is a catenary, unduiary. nodary, ' Vl + jr 5 a straight line parallel to the i-axis, or a semicircle centred on the i-axis. But 2

1 _ dx V l + y' ~ ds 2

3. Surfaces of Revolution with Constant Mean Curvature

so the extrema] equation for our variational problem coincides with that of the roulette of the ellipse or hyperbola ((2.2) and (2.3)).

Rotating each of the roulettes about its axis of rolling produces five types of surfaces in Euclidean 3-space GAUSS MAPS: In an analogy with the case of oriented R . called the surfaces of Delaunay: the catenoids, iwducurves in the plane (§2), we associate to any oriented loids, nodoids, the tight circular cylinders, and the spheres. surface M immetsed in R its Gauss map y : M -» S (the unit 2-sphere centred at the origin in R ), defined by assigning to each point x E M the positive unit VARIATIONAL CHARACTERIZATION: We formu- vector orthogonal to the oriented tangent plane to M late the following isoperimetric principle, for the un- at x. Its differential dy[x) can be interpreted as a symduloid and nodoid (only minor technical changes metric bilinear form on the tangent space TJA. Its eibeing required for the other cases). genvalues k,,X are well determined up to order. The Consider graphs in R of non-negative functions symmetric functions K = k K and H = (\i + are called the curvature of M and the wean curvature of yM*rj,*il— R(^O) the mmersltm at x, respectively. For instance, with fixed volume of revolution (1) the cylinder has K = 0 and constant mean curvature H * 0; 1/(3,) = « (''y'dx: (2) the sphere of radius R has constant curvature K •B = IfR and constant mean curvature H = 1/R; (3) the catenoid has variable curvature K and mean and extremize their lateral area curvature H = 0; (4,5) the unduloid and nodoid have variable curvaAii/) = 2K f 'y ds ture K and constant mean curvature H # 0. 3

3

3

2

J

x

t

2

s

2

2

holding the endpoints fixed. By general principles of constraint (under the heading of Lagrange's method of multipliers for isoperimetric problems [5}), we are led to the Euler-Lagrange equation associated with the integral

F(y) =

»jV

d l+

3

,2

+ 2oyVl + y )dx.

Here o is a convenient real parameter- Its integrand f does not involve x explicitly, so we obtain a first integral from 5*

THE MATHEMATICAL

3

DELAUNAY'S THEOREM: The complete immersed surfaces of revolution in R with constant mean curvature are precisely those obtained by rotating about their axes the roulettes of the conies.

ds)

>o = n ['V

These five surfaces were recognized by Plateau, using soap film experiments. Say that a surface of constant mean curvature in R is complete it it is not part of a larger such surface. From Sturm's variational characterization, we obtain

rtmlXsaKm VOL.

I. NO.

I. iw

Thus Delaunay's surfaces are those surfaces of revolution M in R which are maintained in equilibrium by the pressure of a field of force which acts everywhere orthogonally to M. 3

83 4. Harmonic Gauss Maps

Again, that has an explicit solution in terms of elliptic functions. Furthermore, the associated map y : R* - » S An easy yet vitally important theorem of Ruh-Vilms is doubly periodic, factoring through the torus T <* R IZ [6] states that: to produce a map y : T—- 5. as desired. Incidentally, the integrand of E is A surface M immersed in R* has cuiistanl mean curvature if and only ifils Causs map -y:M->S satisfies the equation ®dyp = a' + y s i n V = \dyfy, Calabi made the beautiful observation that Smith's where A denotes the Laplacian of M with conformal maps y : T —> S are the Gauss maps of certain surfaces of structure induced from that of R', and vertical bars the Delaunay [2j. Euclidean norm at each point. Indeed, (4.1) is the condition for harmonicity of the map y |3j—and is the Euler- A HARMONIC REPRESENTATIVE IN A HOMOLagrange equation associated to the energy (or action) TOPY CLASS: If we represent the torus T in the form integral T = R/nZ x Rj2nZ and use polar coordinates (r,0) on the unit sphere S, then a map from the cylinder to S of the form Uii - U\\dy¥. r •= »(r), » = y M :

1

£ is a conformal invariant of M. SMITH'S MECHANICS: Motivated by certain mechanical analogies, R. T. Smith |7J found solutions to equation (4.1) as maps -y : R —- S, as follows: Think of points of R parametrized by angles (d>,B), and use spherical coordinates on the sphere 5: !

!

e

r

subject to the conditions u>(0) = 0, 4>(n) = n is harmonic if and only iffl>satisfies the pendulum equation (4.3) with A = 1. There are such solutions. Indeed [4], the GAUSS map of the nodoid induces a harmonic map of a Klein bottle y : K -* S. Furf'imnore, that map is not deformable to a cunslitrrl map. Hopf's classification theorem insures that the maps K —* S are partitioned by homotopy into just two classes. Thus the harmonic map y represents the nontrivial class.

2

References ] C- Delaunay, Snr

la surface de revolution dont la courbure moyenne est constants, f. Math, pures et appl. Ser. 1 (6)

If we restrict our attention lo maps y of the special form ,B

(*,fl| = fe sin n(c|>), cos o(o»), then the equation of harmonicity becomes the pendulum equation o" =

j

sin

7. It. T. Smith, Harmonic mappings of spheres. University of

2a.

(4.3)

We assume that a(0) = ir/2, so that the solution oscillates symmetrically about Ttt2. Now a first integral of (4.3) is given by lc - A cos

(1841), 309-320. With a note appended by M. Sturm. 2. J; Eells. On the surfaces of Delaunay and their GAUSS maps. Proc. IV Int. Colloq. Diff. Geo. Santiago de Composltla [1978), 97-116. 3. J. Eells and L. Lemaire. A report on Itarmoiuc maps. Bull. London Math. Soc. 10 (1978), 1-68, 4. J, Eells and L. Lemaire, On the construction of harmonic and holomorphic maps between surfaces. Math. Ann. 252 (1980), 27- 52. 5. A. R. Forsyth, Calculus of variations. Cambridge (1927). 6. E. A. Ruh and ]. Vilms, The tension field of the Gauss map. Trans. Amer. Math Soc. 149 (1970), 569- 573.

2

Warwick Thesis (1972). 0. D'A. W. Thompson, Growth and form. Cambridge (1917). 9. C. Zwikker, TIic advanced geometry of plane curves and their applications. Dover (1963). Mathematics Institute University of Warwick Coventry CV4 7AL England THE MATHEMATICAL

LNIELllCENCEH VOL

9.

NO- ].

IW

57

2

84

B a

manuscripts math. 28, 101 - 108 (1979)

*

2i^£«fi?«i!

mathematica ©ti»Sprjn :cf-Vtrlaiil >''9 1

t

MINIMAL GRAPHS James

Eells

D e d i c a t e d t o Hans Lewy and C h a r l e s B . M o r r e y , J r .

E l e m e n t a r y p r o p e r t i e s of harmonic maps between Riemannian m a n i f o l d s a r e i n t e r p r e t e d v i a t h e i r g r a p h s , viewed as n o n p a r a m e t r i c m i n i m a l s u b m a n i f o l d s (Proposition 1). Then examples a r e g i v e n of n o n p a r a m e t r i c submanifolds of compact R i e m a n n i a n m a n i f o l d s which cannot be deformed - through n o n p a r a m e t r i c submanifolds - to n o n p a r a m e t r i c m i n i m a l s u b m a n i f o l d s ( P r o p o s i t i o n s 2 and 4 ) .

L e t <J>:M and d e f i n e

N be a smooth map between

$:M -+ M * N by $ ( x )

a smooth embedding

= (x,rj>(x)).

of M onto a c l o s e d

M >= N, which we c a l l

the graph

of_ the map ttj •

( N , h ) when we w i s h to

Riemannian c o n s i d e r a t i o n s .

Then $

submanifold

Riemannian m e t r i c s g , h on M,N, we w r i t e form $ : ( M , g ) *

manifolds, is of Given

the map i n

the

emphasise

The graph r

of 4

is

W endowed w i t h the R i e m a n n i a n m e t r i c induced from

that

of

an

(M g N,g x h ) .

e x t r e m a l of

Then

the volume

i s minimal ( i . e . , functional

101

is

EELLS

V(*)

f

=

m

x

1

|A »*( i|dx ...dx

m

M f A [A * Cx)| M p>0 P

=

h e r e m •= dim H , and /^<J> ( x )

is

A

of (1) is

2

f t

J

the d i f f e r e n t i a l

cj> ( x ) )

if

the p -

t h

and o n l y

4: (M,$ (g x h)) -»- (M * N,g x h ) an e x t r e m a l of

E(»)

the energy

=

k\

f f l

dxl...dx

;

exterior if

power

the map

harmonic; i . e . ,

functional

| $ ( x ) | ^ e T ^ ^ g x h ) d x ^ . - d x 'm ' ( x j | - / d e t T (,g x hj d x ' . . . a : 2

A

m

A

' •!

See

[3,

If $ = I x

p.119].

I : M •* M d e n o t e s the so

i d e n t i t y map,

( I x h ;

then thus

we have *

PROPOSITION 1. projections, if

(g

x

h)

-

g + c#> h .

L e t t i n g I T ' , T T " denote the

the map $ i n

the c o m p o s i t i o n s

$'

(1)

indicated

harmonic i f

and o n l y

and 3>" a r e both h a r m o n i c :

102

86 EELLS

(M,g + * h)

(M,g)«f-

(M *

This follows

EXAMPLE 1.

N

,g

x h)

MN,h).

from the c o n s i d e r a t i o n s

[9,6].

m

R , w i t h induced E u c l i d e a n

( N , h ) be

the form $ ( x )

n

R .

= (x,ffi(x))

Then a map $:U -+ R is

nonparametric immersion. k($)

p.133].

L e t (M,g) be an open s u b s e t U of

the E u c l i d e a n space metric; let

in [3,

m + n

of

traditionally called a

Set

= g + ij>*h = $*(g

x h);

thus f o r 1 £ i , j < m we have

where

-

1

B^/Sx .

The t e n s i o n f i e l d s

E u l e r - L a g r a n g e o p e r a t o r s of

the a s s o c i a t e d

f u n c t i o n a l , whose v a n i s h i n g i s h a r m o n i c i t y ; see

l e f t members of the m Z - V ( / d e t lcC
the

energy

the c o n d i t i o n

[ l , 3 ] ) T(4>'), !"($") a r e

by the (2')

(i.e.,

for

represented

systems

1 J

(*))

=

0 ( 1 < j <_ m ) ;

L

(2")

m t i,j=l

(/det ax

1J

k ( » k (H.)

1

=

0.

3

These a r e c l e a r l y e q u i v a l e n t system:

103

to the m i n i m a l s u r f a c e

87

EELLS

m Z -21-1 rJX

(3')

(/det

k(*) k

1 J

($)

=

0 (1 < j

<. m ) ;

1

m E

(3")

=

EXAMPLE 2 .

0.

L e t (M,g) be a r b i t r a r y ,

and take N = R .

Then T ( * ' ) = 0 a u t o m a t i c a l l y i f T<4") = 0; and the c o n d i t i o n T ( $ " ) = 0 means t h a t harmonic f u n c t i o n . our

notation

(M,k($)) + R i s

a

(As we have a l r e a d y e m p h a s i s e d ,

indicates

t h a t $ i s harmonic w i t h

respect

to t h e R i e m a n n i a n m e t r i c k(<J>) on M; i n p a r t i c u l a r , may

depend on

. m Z

In local

it

charts

J

k <•)*..-

=

0,

and

T a k i n g (M,g) to be R , then T(3>) = 0 m . Z ( * . / / l + |d$| ) = 0. i = l 3x

if

2

(4) The

i f and o n l y

equation

form.

(4)

is

quasilinear e l l i p t i c ,

I t i s not uniformly e l l i p t i c ,

the maximum to minimum e i g e n v a l u e s of

the c o e f f i c i e n t s

be

bounded.

of

S.jjJ,

is

104

of

in

divergence

f o r the r a t i o of the m a t r i x ( a 2

1 + |dr)| ,

)

w h i c h may not

EELLS

EXAMPLE 3 .

5

L e t (M,g) and ( N , h ) be K a h i e r

and tf>:M •* N a holomorphic map. n Kahier metric for follows

that *

the complex

manifolds,

Then g + $ h i s a s t r u c t u r e of M.

the i d e n t i t y map

= I

: (M,g + 4> h) *

(M,g) and

*" - <> f : (M,g + 4 h) -

(N,h)

a r e b o t h h a r m o n i c , b e i n g holomorphic maps of manifolds

It

[3, p . 1 1 8 ] .

Kahier

Consequently,

* : (M,g +0 h) * (M x N,g x h) i s h a r m o n i c ; i . e . , any holomorphic map cj>:M -*- N between 11

Kahier manifolds

Let

has m i n i m a l graph T .

(M,g) be the 2 - s p h e r e

S or the r e a l

p l a n e P , w i t h any Riemannian m e t r i c g.

Then f o r

conformal map $iM •* ( N , h ) the m e t r i c g + h conformally equivalent :

to g.

*" = i s harmonic i f case,

is

(M,g)

and

(M,g +
and o n l y i f

(N,h)

(|>:(M,g) •* ( N , h ) i s ;

cf i s w e a k l y c o n f o r m a l , Proposition

and

in

that

is^ a_ m i n i m a l g r a p h .

1 g i v e s an i n t e r p r e t a t i o n of m i n i m a l

graphs i n terms of harmonic maps; g e n e r a l l y ,

in

c o n t e x t the n o t i o n

of h a r m o n i c i t y depends on a

Riemannian m e t r i c

(on the domain) w h i c h i n v o l v e s

map i t s e l f .

any

The i d e n t i t y map

(M,g + *h) -

i s harmonic [ 3 , p . 1 2 6 ] ;

projective

An e x c e p t i o n

a r i s e s i n the c a s e

105

of

that

the

89 6

EELLS

surfaces:

We c a n i n t e r p r e t a r e s u l t

follows:

For

graph

in

keeping

any m e t r i c s g , h

(P * S , g * h )

the same n o t a t i o n

PROPOSITION 2.

is_ h o r i z o n t a l .

[2]

Furthermore,

we have

of

of

the

unique

such maps, t h e n rJ> cannot o

to £i map w i t h m i n i m a l graph

A generalisation

as

o_n P , S e v e r v m i n i m a l

I f $ ;P •* S r e p r e s e n t s

n o n t r i v i a l homotopy c l a s s be deformed

in [2,7]

i n P x S.

a theorem of Lawson [5^

gives

PROPOSITION 3 .

E v e r y m i n i m a l graph of p x S

3

is

horizontal.

Here S

d e n o t e s the E u c l i d e a n n - s p h e r e .

contrast, mal

it

graphs

By way of

i s w e l l known t h a t t h e r e a r e many m i n i -

in P x s

n

f o r n >_ 4, which a r e not

horizontal.

We can i n t e r p r e t the Proposition

2,

theorem i n

[4]

as we d i d

t a k i n g i n t o a c c o u n t Example 3:

106

in

90

EELLS

Let

( T , g ) be the

PROPOSITION 4.

2-torus,

7

w i t h any R i e m a n n i a n m e t r i c

The graph r

o f a harmonic map

4>: ( T , g ) •+ ( S , h ) of_ s t r i c t l y p o s i t i v e submanifold

g.

degree i s _a K a h i e r

(hence m i n i m a l ) of T x S .

No map :T + S o f o

degree 1 _is_ homotopic to a. map

w i t h m i n i m a l graph i n T * S.

Finally, be adapted

let

to the

us note t h a t

can

c a s e of harmonic s e c t i o n s $ o f

Riemannian f i b r e b u n d l e s map

these constructions

TT:W •* M, g e n e r a l i s i n g

the

(1).

REFERENCES [l]

E E L L S , J . and L . LEMAIRE, A r e p o r t on harmonic maps. B u l l . London Math. S o c . 10, 1-68 (1978)

[2|

E E L L S , J . and L . LEMAIRE, On the c o n s t r u c t i o n of harmonic and holomorphic maps between s u r f a c e s . (To appear )

[3]

E E L L S , J . and J . H . SAMPSON, Harmonic mappings of Riemannian m a n i f o l d s . Am. J . Math. 86, 109-160 (1964)

[4]

E E L L S , J . and J . C . WOOD, R e s t r i c t i o n s on harmonic maps of s u r f a c e s . Topology 15, 263-266 (1976)

107

91 EELLS

[5]

LAWSON, H . B . , Complete m i n i m a l Ann. Math. 9 2 , 335-374 (1970)

[6]

LAWSON, H. B . and R . OSSERMAN, N o n - e x i s t e n c e , n o n - u n i q u e n e s s and i r r e g u l a r i t y of s o l u t i o n s to the m i n i m a l s u r f a c e s y s t e m . A c t a Math. 139, 1-17 ( 1 9 7 7 )

[7]

LEMAIRE, L . , On the e x i s t e n c e of harmonic U n i v e r s i t y of Warwick T h e s i s (1977)

[8]

OSSERMAN, R . , M i n i m a l v a r i e t i e s . Math. S o c . 7 5 , 1092-1120 (1969)

[9]

OSSERMAN, R . , A s u r v e y of minimal s u r f a c e s . Van N o s t r a n d Math. S t u d i e s 25 (1969)

[10]

surfaces in

Bull.

3

S .

maps.

Amer.

OSSERMAN, R . , Some p r o p e r t i e s of s o l u t i o n s to the minimal s u r f a c e system for a r b i t r a r y codimension. P r o c . Symp. Pure Math. Amer. Math. Soc. 15, 283-291 (1970)

James E e l l s Mathematics I n s t i t u t e U n i v e r s i t y of Warwick Coventry, England

(Received January 2o,

108

1979)

92

Mathematische AlUISiCII

Math. Ann. 252, 27-52 (1980)

© by Spnnger-Vcilag 1980

On the Construction of Harmonic and Holomorphic Maps Between Surfaces* 1

J. Eells and L. Lemaire

2

1 Department of Mathematics, University of Warwick, Coventry C V 4 7AL, U K 2 Department de Mathematiqtte, Universite Libre de Bruxelles, Boulevard du Triomphe, B-1050 Bruxelles, Belgique

Introduction (A) Let M and N be compact surfaces without boundary; and a homotopy class of maps of M into N. We shail denote by & the integer which is the common value of the twisted degree of any map <^eJff; and the index of the subgroup (b^n^M)) of TT.IJV). The following result is an immediate consequence of recent work of Edmonds [14]; see Theorem 1.3 below: x

Theorem 1. Suppose that M and N are orientable, and that Jff is a homotopy class with d 4=0. For any complex structure on N there is a complex structure on M relative to which JV contains a holomorphic representative if and only if \d .\>j , or 4>^:%^Mj-tn^N) is injective. x

x

x

(B) Ifg and /tare Riemannian metrics on Mand N, thenamap<^:M-*N [which we will often write tp:(M,g)-*(N,h) for emphasis] is harmonic if the divergence of its differential vanishes everywhere; see Sect. 2 for details. We let S,T,P,K denote the sphere, torus, projective plane, Klein bottle, respectively. In the present paper we will establish Theorem 2. The surfaces M and N admit Riemannian metrics relative to which 3nf contains a harmonic representative, except 1) when Af= T, N = S, and \dj = \; 2) when M — T,N — P, and y? is the class obtained from that in I) by composition with the covering map n:S~*P; 3) when M = P, N = S, and jj? is the nontrivial class; 4) when M — P,N = P, and j>f is the class obtained from that in 3) by composition with n; 5) perhaps when N — P and for certain classes of nonorientable maps M->P with mod 2 degree = 0. (That case does not arise. See [28, §5.8J.) * The main results in this paper were presented at the Mathematische Arbeitstagung (Universitat Bonn) in June 1978

0025-5831/80/0252/0027/S05.20

93 J. Eells and L . Lemaire

We know of no case where 5| provides a new exception. Of course, Theorem 2 bears on the fundamental Problem [15]. Given Riemannian metrics g, h on M, N, when does harmonic map :{M,g)-r{N,h)'>

contain a

It is known [see Corollary (2.7) below] that the answer is always, provided + S orP; this implies the corresponding statement in Theorem 2. Of course, it also follows as an application of the main existence theorem in [15]. There will be several other circumstances in which we can provide a solution to the problem; for instance, if M and N are spheres or projective planes; see Sect. 3 below. The cases 1) and 2) were established in [17]. Cases 3) and 4) were found by the authors - and first appeared in [27]; see Sect. 3 below. The case N = S was treated in [29] for M orientable; and is Theorem 4.2 for M non orientable. (C) Our presentation is organised as follows: In Sect. 1 we recall the homotopy classification of maps between surfaces, and formulate Edmonds' theorem characterising those homotopy classes which contain branched immersions. Section 2 summarises the properties required of harmonic maps of surfaces. In Sect. 3 we obtain a complete description of harmonic maps between spheres and projective planes, with arbitrary Riemannian metrics. Section 4 is devoted to the existence part of Theorem 2, when IV=S; and Sect. 5 to that when N = P. In Sect. 6 we give examples of harmonic maps M-*S which arise as Gauss maps of immersions of M in R with constant mean curvature, especially in relationship to the examples of Lawson. Section 7 develops some related constructions for such immersions in IR ; we aiso adapt a geometric interpretation (due to Calabi) as Gauss maps of an analytical construction (due to Smith) of harmonic maps 4

3

T->S. K These examples have served as primary motivation and inspiration to us, in our efforts to understand Theorem 2. Finally, in Sect. 8 we discuss briefly several related constructions of harmonic maps of surfaces into certain higher dimensional homogeneous spaces. I . Homotopy Classification of Maps Between Surfaces Any compact connected surface A/{ + S) without boundary is differentiably the connected sum ( # ) of projective planes and tori; accordingly, its fundamental group 7E,(M) is presented by generators {a^ g ; „ (b ,c ) ^ ^ subject to the single relation {J7 a?} (njlbjcpj cJ '))= 1. If M is orientable, it is a sum of p tori, where p = genus(M); thus there are no u/s in the presentation of 7t,(M). If M is nonorientable, it admits different representations; for instance, T#P=P#P#P; we choose to realise M as a sum ofk projective planes, k is called the genus of M, and %(M) is generated by ( a ^ , < subject to the relation TIaf=l. For any two compact connected smooth surfaces M, N without boundary, consider the space ^(M,N) of maps M-*N; these can be of any class C" for Ogn g

l

(

ifii

J

J

l

j

m

94 Harmonic and Holomorphic Maps Between Surfaces

29

goo, unless otherwise specified. The components of ^{M,N) are the homotopy classes, the set of which we denote by [Af, JV]. The following enumeration [in (A)] and classification [in (B)] results on homotopy classes are well known: (A) Enumeration of \_M,N~} Case N + S or P [21]. Two homomorphisms 0, 0':7t (M)--7r (JV) are said to be conjugate (8^6') if there is an element ben,(N) such that 6'{a) = b~ 6(a)b for all aen^M). Any continuous map M-*N induces a conjugacy class of homomorphisms from rc (M) to n^N). When N + S or P, these conjugacy classes are canonically identified with the components of '4{M, N): 1

1

l

1

mm

(1.1)

*

HoMnJMU^N))/^)).

Case N = S [8]. Then the assignment tp-'d^, ( = the degree or degree mod2 of
[M,S~\^H\M;1)=%

if hi is orientable, — 2l if Af is nonorientable. 2

CaserV = P [ 8 , 34]. Then 1

Hom(7r, (Af), n^PM ») = Hom(7r, (Af), E ) = H (Af; J, ). 2

2

Using that identification we say that a homomorphism 0 ;it {M)-*7i (P) is oriented if 0 = Wj(A/) (=the first Stiefel-Whitney class of Af); and nonorientable otherwise. Geometrically, 0 is oriented if the maps d> which induce it are oriented; i.e., if when X is an orientation-preserving (-reversing) loop of Af, then d>°k is an orientationpreserving (-reversing) loop of N. Equivalently, let S'p-^P denote the orientation sheaf of P; then 0 is oriented iff Q~ 3~ = 3~ . The classification of [Al, P] is thus partitioned into two subclasses depending on the orientability of the induced homomorphisms. Note that being given 0 is equivalent to being given a homotopy class M->P" for any n £ 3 ; and we are asked to classify those elements 0 of [Af, P"] which have image in [Af, P] under the inclusion map P-»P". l

l

1

f

M

a) 0 is oriented. Then 6 is represented by infinitely many homotopy classes: these are parametrised by the absolute value of the twisted degree d. It takes the values t/s0mod2 if Af is orientable; t/s genus (Af) mod 2 if M is nonorientable. b) 0 Is nonorientable. Then 0 is represented either by 1 or by 2 homotopy classes in [Af, P ] , as follows: If the degree mod2 d (ff)=l, then we have 1 class (equivalently, Sq 0 + 0, as Epstein has noted); rfj(0) = O, then we have 2 classes (equivalently, Sq' 0 = 0); these being parametrised by H (M,8-'3r ) = z. . 1

2

2

p

2

(B) Classification of\M,N'\. The only case not covered in (A) is that of b) with d 6 = 0. We must determine how to distinguish those two cases. 2

95 30

J . Eells and L . Lemaire

One method is as follows [4]: Given two maps

^ d> rel. x iff d E Image a , where a is a homomorphism 0

0

1

M i

2

a

0

i

0

M i

9

9

(in the notation of [4]) defined via the boundary operator in the homotopy exact sequence of the fibration ^(Af, P J - ^ A f ' ^ P ) defined by restriction. In the case under consideration, we know [34] that there are two classes so that image a = 2ZC? _ and the homotopy classes relative to x representing 8 are in bijective correspondence with coker a = Z . Olum's description [34, 33, 35] of the classes determining 8 is as follows: Subdivide Af into a simplicial complex with ordered vertices (x,|, with x as first vertex. For each i choose a path U from x to x and for any ordered 1-simplex x Xj define y^^K^M^a) by VfajX^V]'. Construct the simplicial map g :M-> ; define a on Af by 7

9

0

fl

2

0

j

0

f

t

iOI

g

0

a

111

(x x )-»y (

y

if

0 !

(x,.X;HP< >

6(y{x x )) = e i

if

]

0

OiyiXfX^e,.

1

Here P " is a projective line in P representing e EiiA.P,y ); e is the neutral element of n {P,y^\. If (XfXjXj is an ordered 2-simplex and i

0

0

Y

111

,L

then define a to be a homeomorphism onto P —P . Otherwise, map Then g represents 8, and d (g ) — d {8). Finally, define the map

{x x x )-'P '.

9

e

2

e

i

j

y

2

p-.n^y^HHM^-'^,,)

by

/f(e-,) = obstmction to deforming g to g with x traversing P description of /J(e,): Assign - 1 to (XjXjX,) if B

e

0

l n

[Alternative

9(v(x,x ))- =%(x xJ); ;

ei

;

and 0 otherwise. I.e., /?(e,) = 0if an even number of — 1 are assigned; and /!(£,)= 1 if an odd number of —1 are assigned.] Then j3 is a crossed homomorphism pVw') = $w) + w<./?(*"), 2

l

where «denotes the action ofn,(P,y )on H (M ;8~ 3~ ); consequently, it partitions H^M-J-'Pp), the orbit of h being 0

C(fc) = { ^ ) + ^ ^ : w

e f r i

P

(P,y )}. 0

Thus C(0)={0,/J( )}, e(l)={l,P(e )+e ol}. ei

Case / ^ H O .

T

h

e

n

1

1

C(0)={0}, C(I)={1}. This is the case deg g = 0. 2

9

31

Harmonic and Holomorphic Maps Between Surfaces

Case /J(e,)=l. Then C(0)= (0,1} = C{1). This is the case deg g„ = l . 2

(C) Asurjective map d>:M->N between surfaces is a branched cover if it is finite-toone and open. A branched cover is an oriented map, in the sense that it preserves the I ' Stiefel-Whitney classes: >*w (/V)=w (M). s

J

9

1

1

The following is well known; if N is not S or P and d>\M->N is a branched cover, then its induced homomorphism tp^ :n: (M)-'n (N) is injective iff 0 is unramified; i.e., is a covering map. l

l

Indeed, the sufficiency is standard in the theory of covering spaces. For the necessity, let 9:N-*N be the unramified covering associated to the subgroup 4>^n,(M)) of jt,(rV> Then is therefore of degree +1 and being a branched covering is itself a diffeomorphism. It follows that

N with degree +0 is homotopic to a branched cover iff (1) ^ ( N H w ^ A f ) ; and 2) \dj>j^ or tp^m^M^K^N)

is injective.

Here \d^\ is the absolute degree. As the notation suggests, jd^j is the absolute value of the usual degree when Af and N are orientable. It has been shown recently [6] that if Af and A/are orientable with genus (Af)^d •genus (AOfor some integer
x

+

1

1

2. Harmonic Maps of Surfaces Background references: see [15, 16]. If
m m = \ m f ^

where the vertical bars indicate the Hilbert-Schmidt norm of the linear map (the differential of

(2.2) Bm=Smm&-

)

97 32

J. Eells and L . Lemaire

A smooth map 4>e^{M, N) is harmonic if it is an extremal of E. Thus it satisfies the Euler-Lagrange equation (2-3)

T<0)=O,

where x(cp) = Trace Vdd, is the tension of <j>. Suppose that M and N are surfaces. Then we can introduce isothermal charts relative to the Riemannian metrics g and h. Calling z, z (resp. w, w) the complex variable and its conjugate in these charts, we can write g = Q (z)dzdz, h = G {w)dwdw. Then 2

(2.4)

£(0)=,|rj»[|wj * M

2

1

3

+ |vv | ]^i, 5

In particular, we note that i / M and N are orientable and tp :M-*. N is holomorphic or antiholomorphic relative to the complex structures determined by g and h, then

*h = ug. It is known that the (2,0)-part of the quadratic differential / is weakly conformal. A nonconstant weakly conformal harmonic map

— 0; around the image of each of these points there are normal coordinates in which
4>\ ) Z

C

Re(z") + (|z|'); 0

c l m ^ + oflzl*); oi\z\*) for

aS3;

see [19] for further details. If both M and N are surfaces, then a branched immersion is a branched covering. // cp is harmonic and its Jacobian J ^ S O on M, then

) for any map d>:M^N,and tp°\p is harmonic iff (p is harmonic. In particular, the energy (2.4) and the notion of harmonicity are conformal invariants of (M,g). More generally, the composition :{M*g)->{N,h) inducing 0, minimising E amongst all maps representing 6. s

98

Harmonic and Holomorphic Maps Between Surfaces

(2.7) Corollary. If

K (N)=0 2

(i.e., JV* S or P) then every homotopy class Jt?e [Af, jV]

has a harmonic representative

cp:(M,g)-*(N,h).

3. Maps Between Spheres and Projective Planes When Af and N are spheres or projective planes, equipped with any given Riemannian metrics, we have a complete description of all harmonic maps.

(A) Maps from Sphere to Sphere

First we recall the description of the harmonic maps in question [28, 42]: (S, h). Any such map is conformal, or equivalently + holomorphic (i.e., holomorphic or anti-holomorphic) with respect to the associated complex structures on Af and N. Thus in order to find all such harmonic maps, we can represent both copies of the sphere minus one point as complex planes. Then a harmonic map

can be represented by the rational functions (a , b e C) t

3

or

If these quotients are irreducible and a, #0, bj+0, then the degree of such a map is + maxfr, s). All degrees are thereby represented.

(B) Maps from Projective Plane to Sphere

There are two homotopy classes of maps from P to S, one of which contains the constant maps. (3.3) Proposition. Any harmonic map d>:(P,g)~-(S,h) is constant. In particular, the nontrivial homotopy class does not contain a harmonic representative.

Proof. Let tp be harmonic. Composition of with the covering map p:S-*P harmonic map cj>:

gives a

(S,p*g)

(P,0)-*> (S,h) 3

2

The degree of d> is 0, for its induced homomorphism tp*:H (S)->/f (S) factors through H (P)=Z . Since any harmonic map {S,p*g)-*(S,h) has the form (3.1) or (3.2), we conclude that must be constant. 2

2

99 J. Eells and L. Lemaire (C) Maps from Sphere to Projective Plane Any map tp :S-»P lifts to a map 0: (S,p*h) (S,g)

(P,h)-

The twisted degree of

; and because

in such a homotopy class lifts to a map
1

1

(S,p*g) 4 {S,p*h) (P,g)

$

(P,h).

Since rb is harmonic it is conformal, together with
wi-lzy^-Wz))- . If

&,=(-!)'5,

[We do not have to consider maps given by (3.2), for they project on the same class of 0's.] All classes are represented harmonically (for instance, by the maps w — z ); and we obtain an explicit description of every harmonic map. The second family, associated to the zero morphism $:n (P)-*n {P), contains two homotopy classes, including the trivial one. All maps in these classes lift to maps from P to .9, and any such harmonic map is constant [as shown in (3.3)]. In summary: 4

1

l

(3.5) Proposition. There are two families of homotopy classes of maps from (P,g) to (P, h). The first contains an infinite number of classes, all of which contain harmonic

100 Harmonic and Hoiomorphic Maps Between Surfaces

35

representatives. These are given by (3.1), subject to condition (3.4). The second contains two homotopy classes: The trivial class, in which every harmonic map is constant; and a class which contains no harmonic representative. 4. Maps of a Surface to a Sphere When M is not a sphere or a projective plane, we have only partial existence results; therefore we shall concentrate on the following question [28]: Given two compact surfaces Atf and N and a homotopy class 3^ of maps from M to N, does contain a map which can be rendered harmonic by a suitable choice of Riemannian metrics on M and N ? In other words, do there exist metrics on M and N such that #f contains a map harmonic with respect to these metrics? (A) Maps from an Orientable Surface to the Sphere For completeness, we first recall the known results concerning the harmonic maps from an orientable surface to the sphere; let p — genusM. The homotopy classes of maps M-»S are parametrised by the degree d of the maps, and it is shown in [17] that for \d\ ^p any harmonic map of degree d is + holomorphic. Such maps exist in the following cases: if \d\^p+l; if \d\ = p and M is nonhyperelliptic (i.e., the associated Riemann surface is not a holomorphic 2-sheeted branched covering of S); if \d\ = p is even. On the other hand, when \d\ = p is odd and M is hyperelliptic, there is no + holomorphic map of degree d from M to S; and hence no harmonic map in these classes. For the remaining degrees, it is known [26,28,29] that if Af is a surface of genus p and \d\ g p — 1, there exist metrics on M and S such that there is a harmonic non + holomorphic map from M to S of degree d. This map is not an absolute minimum of the energy in the class. These results provide the following answer to the rendering problem [28]: (4.1) Proposition. Let Mbea surface ofgenus p and d an integer. There exist metrics g and honM and S and a harmonic map d> :(M,g)^{S,h) of degree d in all cases except when p— \ and \d\ — 1. (B) Maps from a Nonorientable Surface to the Sphere In contrast to (3.3): (4.2) Theorem. Let Mbea nonorientable surface of genus k S: 2 and S a sphere. There exist metrics on M and S such that the (unique) nontrivial homotopy class of maps from MtoS contains a harmonic representative. As above, this map is not an absolute minimum of the energy in that class. Proof. The difTerentiable manifold M is a connected sum of k projective planes; we shall represent it as a connected sum M of hemispheres, quotiented by the identification of antipodal points on each boundary circle L,(t—1,k).

101 36

J . Eells and L. Lemaire

We equip Af with a metric inducing a smooth metric on Af such that Af appears as a surface in R symmetric with respect to k + 1 planes, one of them orthogonal to the k others; and these forming equal angles. We call („,/;,...,/,, the lines of intersection of these planes with Af and S (s = 0, ...,fe) the symmetry with respect to / (see Fig. 1, where / is in the plane of the paper). On S we choose the metric of a surface of revolution symmetric with respect to its equator (for example, the Euclidean metric). We call the equator /'„ and choose k meridians forming equal angles, put in correspondence with the l and called t' (see Fig. 1). We denote by S' the symmetry with respect to Finally, we associate to each boundary circle L , a point P, on S as indicated in Fig. 1. 3

s

s

0

3

s

s

We note that if k is even and k>2 it would be sufficient for the construction below to impose symmetry with respect to k/2+1 lines (see Fig. 2 for k = 4):

Fig. 1

Adapting the method of [28, Sect. 11], we shall use the direct method of the calculus of variations to find a harmonic map of degree one 0:Af-»S, equivariant with respect to the groups of symmetries, and mapping L , on P,. Then we shall check that 4> induces a C -harmonic map from Af to S. i) Following [32, Sect. 9.4], we say that a measurable map
2

J

102 Harmonic and Holomorphic Maps Between Surfaces

37

ii) Let us say that a map 0 :M-»S satisfying rpS = S' <j>(s = 0 k) is equivariant. Consider a minimising sequence {0 } for the energy functional E in the class of equivariant L^-maps sending L , to P,. (An Z?i-map has an L trace on L,.) First of all, we observe that we can replace {0 } by another minimising sequence {4>' } in the same class, where 0J has the additional property of mapping (almost everywhere) a region of M bounded by three lines of symmetry and a piece of L , in the corresponding region of S (see Fig. 3): s

s

f

1

r

r

ill

Fig. 3

For that it suffices to associate to each
r

r

2

2

r

v

2

r

2

£(0)£liminfE(0;) [32, Lemmas 9.4.16 and-9.4.15]. iv) In each region, 0 minimises E with respect to the Dirichlet problem induced by its trace on the boundary. By a fundamental theorem of Morrey ([31] or [32, Theorem 9.4.2]) 0 is C™ in the interior of the region. v) We observe that the method of [31, 32] enables us to prove that 0 is continuous on each line of symmetry, away from the intersections with the other lines or the Z.,'s. Indeed, for two sufficiently nearby points on such a line, the proof will involve only discs centred on the image of the line in an exponential chart; and the fact that 0 minimises E in such a disc amongst the equivariant maps is sufficient for us to proceed with the proof (see [28, Sect. 12] for details).

103 38

J. Eells and L. Lemaire

On each line L„ $ is continuous by construction, for it has constant image P,. By the boundary statement of [32, Theorem 9.4.2], 4> ' therefore continuous everywhere, except perhaps at the intersections of two lines of symmetry or one line of symmetry and one L , . vi) By identification of the antipodal points on the L,'s, $ induces a map d>:M-*S with the same continuity properties. In a small enough disc centred at a point on a line of symmetry, we know by construction that

t

r

0

,p (x,y)=-rp°(-x y). !

Conversely, a map d> satisfying these relations also satisfies d>"(0,y)—0; i.e., <j> is obtained from a map 0of a neighbourhood of a point of L , to S such that 0(Z,,) - P,. A minimum of E for the given Dirichlet problem is therefore a minimum of E in the class of equivariant maps for these new relations. vii) Using this, we now show that (except at the intersection points) <j> is weakly harmonic; i.e., d> satisfies x(4>) — 0 in a weak sense. If D is an open neighbourhood of a point such that D does not contain a point of intersection,

X + ^X

X-£r*X

=X,+X ; 2

P

thus SfX =X and S X =-X . Then V E(d>) = 0, because X , induces a variation in the class of equivariant maps and

l

2

2

Xi

= F_ E(0)--f £(0), K2

X)

hence V E{d>) = 0. viii) Except perhaps at the points of intersection of the symmetry lines, tb's weakly harmonic and continuous. Therefore it is C " [41]. By a theorem [37] on removable singularities, ' C™ on the whole of Af. By construction (and in particular because each region is sent into the corresponding region), tp has degree 1 mod 2. x

s

104

Harmonic and Holomorphic Maps Between Surfaces

39

5. Maps of a Surface to a Projective Plane (A) Maps from an Orientable Surface M-*P Consider first the zero morphism din^Mj-m^P). It is oriented, and therefore is induced from an infinite number of homotopy classes. Any map 0 inducing 8 lifts to a map 0: S

V M

r

1 P

The degree of 0 is twice that of 0, and the nonnegative even numbers parametrise the possible homotopy classes. (The classes of maps 0 of degrees k and - k project to the same class with absolute value of the degree 2k.) All statements made about the maps from M to S in Sect. 4 A apply therefore to the maps from M to P (the notion of oriented holomorphy being replaced by that of conformality). Suppose now that 8 is a nonoriented morphism. It is induced by two homotopy classes, and Theorem (2.6) shows that one of them contains a harmonic map, which minimises E in the union of the two. Unfortunately, that general statement does not indicate which class is represented. Independently we have the (5.1) Proposition. One of the classes associated to 8 can be represented by a harmonic map whose image is a geodesic. Proof. The morphism 8 is characterised by specifying the images of the generators {b„c^ of it,(Af) in B (P) = Z = {* ,e }. i

2

0

I

We can define a morphism from jr (Af) to 2 by mapping the generators sent to e onto 0 and those sent to e onto 1. The relation nbfifc 'c " = 1 is preserved by that map, which therefore induces a homomorphism. Its composition with the morphism from 2 to 2 , defined by (0, l)-*(e , t ) is the given homomorphism 8. To the morphism 7t,(M)-*Z is associated a homotopy class of maps from M to S , which contains a harmonic representative. Its composition with a map from S' to a non con tract ible geodesic of P is harmonic and induces the homomorphism 8. We do not know whether this geodesic minimises E in the union of the two classes associated to 8. It is not excluded that the answer to that question would depend on the metric of M. Concerning the other homotopy class associated to 0, we have results only for certain morphisms. For instance, if 8 is defined by mapping every b to e and every c, toe,,a construction using symmetries as in Sect. 4B yields a harmonic map for certain metrics, as indicated on Fig. 4. (Here the genus is 3 and the boundary circle L, is identified with I*.) An equivariant map is built from the surface with boundary to S mapping the boundary circle L,(L*) onto P,(P*). It factors as a map 0 from M to P, inducing the given morphism. One verifies easily that the separation element of a suitably chosen map homotopic to 0 and a harmonic map inducing the same morphism and whose image is ageodesic.is the generator of n (P). (l.B) implies then that these two maps are not homotopic (free and based homotopy coincide in this case). t

0

1

l

;

e

3

0

1

t

2

0

105 40

J. Eells and L. Lemaire

Fig. 4

L',

This construction is valid for any genus M = p>0. When studying the rendering problem, we can compose the harmonic map ,'s to e . If p + 2, a similar construction permits us to represent the class associated to a morphism mapping exactly onec,. t o e This is illustrated in the case p = 3 in Fig. 5: A curve from one boundary of the cylinder with handles to the other represents that c , and the handles are placed symmetrically around the cylinder (which requirement imposes the restriction on p). ;

(

t

r

t

L

Fig. 5

Variants for these constructions allow us to treat other homomorphisms, but do not seem to lead to a complete solution to the rendering problem in all classes in [M,P]. (B) Maps from a Nonorientable Surface M->P: Oriented Homomorphisms Let M b e a nonorientable surface of genus k^2 and 8:n (M)->n lP) the oriented morphism. To 8 is associated an infinite number of homotopy classes of maps M-*P, parametrised by the absolute value of the twisted degree, which takes all even (resp., odd) values if k is even (resp., odd). l

1

(5.2) Proposition. For any homotopy class of maps from M to P determining the oriented morphism, there exist metrics on M and P such that the class contains a harmonic map. Except when the degree is 1 the metric on P can be prescribed.

41

Harmonic and Holomorphic Maps Between Surfaces

Proof. First of all, we treat the case of degrees > 1 and prove that for all k and d 2: 2 there is a d-sheeted branched covering of P by a surface of genus k. Since the projection map of any branched covering is oriented [14, Theorem 5:2], these maps are in the prescribed homotopy classes. For a given metric on P, they can be made conformal (hence hannonic) by the choice of suitable conformal structures on M. We shall have to use two different constructions, according to the parities of k and d. i) d — 2a^2. We consider two copies of (P,h) and a—\ copies of S, viewed as Riemannian universal covering of {P,h}. We glue these copies together along s segments, s^a. Thus we cut the two first copies along the same segment (i.e., two segments such that one is the copy of the other) and join them crosswise along the slit. Then we adjoin the other copies, one at a time, using different segments; and attach the last one along the s - a +1 remaining segments. This defines a branched covering 0 : M — P which can be lifted to a branched covering of the oriented two sheeted coverings: M

s

£•

I M 4

I P.

0 is a d-sheeted branched covering of S by M , obtained by joining copies of S along 2s segments. Its ramification index f is hence 4s. Using Hurwitz' formula X(M) + r = 2a (S) X

(where x denotes the Euler characteristic) we obtain the genus of M, denoted by y(M)

k:2-k

= x(M)= ^ y ^ = 2 o - 2 s ; k=2s + 2-2a. Therefore k takes all positive

even values. Conversely, for any such k we obtain any positive value of a, and hence represent all classes associated to a positive degree (necessarily even). ii) d = 2a +1 £ 3. We repeat the construction, this time using a copies of S and one copy of P; for the genus of the branched covering we obtain k = 2s +1 - 2a. Thus k takes all positive odd values; and for any such k we get all positive values of a and represent all classes associated to a (necessarily) odd degree S 3. iii) Degree 0. k must be even and the class of maps of degree 0 can be represented by a harmonic map whose image is a closed geodesic. Indeed, let [a^ , be a set of generators of T C ( M ) , subject to the relation l=

t

1

naf=i We define a homomorphism from 7i,(M) to Z by mapping half the a,'s to 1 and the others to - 1 . Therefore the relation is preserved. The composition of this morphism with that from Z to Z defined by (0, l)-»(e , f.^) maps all a,'s to e,, and hence is oriented. Thus we ohtain the harmonic map as in the proof of Proposition (5.1). 2

0

iv) Degree i. (This case cannot be treated as in ii), for a branched covering of degree 1 would be a diffeomorphism.)

107 42

J. Eells and L. Lemaire

The two-sheeted orientable covering M of M is of even genus p = k — 1. As in Fig. 6 let us equip M with a metric symmetric with respect to three planes and S with a metric possessing the same symmetries (for instance the Euclidean metric):

Kg. 6

Using the same technique as in Sect. 4B, it was shown in [26, 28] that there is a harmonic map of degree one from M to S, equivariant with respect to the groups of symmetries. The line I, is sent onto l\ and the equivariance with respect to S S and S' S' implies that antipodal points of /, are mapped to antipodal points of f By identification of antipodal points, we obtain a harmonic map from a surface to P. Because this surface contains a Moebius strip, it is nonorientable and hence of genus k. 0

0

2

2

v

(C) Maps from a Nonorientable Surface A/->P: Nonoriented homomorphisms Let M be a nonorientable surface whose fundamental group is generated by =!

*•

(5.3) Proposition. Let 8 be a nonoriented homomorphism from JI,(M) to 7r,(P), mapping an odd number of the a,'s toe . There is precisely one homotopy class whose maps induce 8, and it contains a harmonic map. i

This is an immediate consequence of Theorem (1.1). Note that these maps do not factor through the circle as geodesies, because they must be of degree d = 1. When 8 maps an even number of a 's on e „ it is induced by two distinct homotopy classes. As in Sect. 3C we see that one of these classes contains a harmonic map minimising E in the union of the two classes; one of the classes contains a harmonic map whose image is a geodesic. 2

f

In the special case of the zero homomorphism, the associated maps lift to maps from M to S; and thus for certain metrics, both classes are represented, by Theorem (4-2). For certain other morphisms, constructions as in Sect. 5A give representatives in both classes; but again this does not seem to lead to a complete solution to the rendering problem.

43

Hannonic and Holomorphic Maps Between Surfaces

(D) Example. Maps from the Klein bottle K-*P The fundamental group of K is generated by the elements a and a , subject to the relation a\a\ — 1. There are four homomorphisms from n^M) to n,(P), determined by the images of a and a . We have the following existence properties for these homomorphisms, thus solving the rendering problem: i

1

a

a

e

2

2

e

i) ( i, )~*( o> or- There are two homotopy classes associated with this morphism, composed of maps which lift to S. One of them contains the constant maps (which are harmonic). The other contains a harmonic map for certain metrics, deduced from a remark of Sect. 5C or from the explicit construction of Sect. 7B below. 2

(ii) (a a )^>(e ,e )or (e e ). There is only one homotopy class associated to each of these morphisms, and it always contains harmonic map by Proposition (5.3). v

2

0

l

v

0

We remark that if P carries the Euclidean metric (induced from its universal cover Sj, then such a map can be built explicitly (see Sect. 7B below). iii) (a ,a )-*(e ,e ). To this oriented morphism is associated an infinite number of homotopy classes of maps K->P; these are parametrised by the absolute value of the degree, taking only even values. l

2

1

1

For ti=0, the homotopy class can always be represented by a geodesic. For
6. Examples of Harmonic Gauss Maps Certain homotopy classes in [M, S] can be rendered harmonic as Gauss maps, as follows: The Grassmannian of oriented planes through the origin in R is the homogeneous space a

G ° = SO(4)/SO(2)xSO(2). 2

As a projective variety it is the complex quadric. It can also be described [12] as 2

4

G%_ = {xeA ]t\ :W\ 2

1

= l

and

OIACC-O}. Z

4

The de Rham-Hodge star operator on 2-vectors is an involution (setting A = A R )

109 44

J . Eells and L . Lemaire

with eigenspaces A corresponding to the eigenvalues ± 1 ; the oriented plane ae G° is sent by * into its oriented orlhogonal complement *a. Thus we have the orthogonal direct sum decomposition A=A ®A_ into 3-dimensional Euclidean subspaces. If we decompose each a€G^_ into the indicated form a = a +a_, then ot eS , the unit spheres in A . Thus we have a diffeomorphism ft :Gj{ - > S x S_ defined by fi(a) = (a ,a_). If ( e ) - is an orthonormal base for fR and we set e — e Ae then (e + e )/\/2, {e,i + e )f^2, (e + e )/}/2 is an orthonormal base for A . Let us note in passing that ±

2

+

+

2

±

±

±

2

+

4

+

k

p

;

12

1Sl

S4

i4

tJ

2A

ia

2i

±

f i ° * ° f c ( a , a _ ) = ( « , — ct_). _1

+

+

In fact, h is an isometric biholomorphic equivalence between these two EinsteinKahler symmetric spaces; see [24; Chap. XI, Exercise 10.6]. We note also that S x S_ is naturally minimally embedded in the sphere S( |/2) of radius \/2 in A; thus we find G° embedded in A with constant mean curvature. +

2

4

(6.1) Let M be a closed oriented surface embedded in 1R , with Gaussmapy .M — G\ . If c eH (G° _ \7L) are the images under h* of the generators in H {S ;Z), then 2

1

2

±

2

(6.2)

±

y*( )[Af> (Af)/2, C±

2

where x(M) is the Euler characteristic of M. That is a theorem of Blaschke [7]; we sketch the proof given in [12]: Let CBH (G% ; Z ) be the Chern class of the canonical circle bundle over G\ . Then c = c + c_. If y is the Gauss tangent map, then *°y is the Gauss normal map, and y*c and (* °y)*c are the Euler classes of the tangent and normal bundles of M in fR . Then ;t(M) = (y*c) [ M ] = ( y * c ) [M]+(y*c_) [A/]. On the other hand, since Af is embedded in IR , 0=(*°y)*c[A/] = ( j * c ) [ M ] - ( y * c _ ) [A/]. The equation (6.1) follows. Now we apply that result to certain of Lawson's surfaces [25]. For every pair of integers m^k^l he has constructed an oriented surface gLj of gen us mk with Euler characteristic j,) = 2(l -mk), minimally embedded in S . If y is the composition 2

2

2

+

4

t

4

+

3

±

where n denotes the indicated projection, then (6.1) shows that deg(y )= 1 -mk. A theorem of Ruh and Vilms [36] asserts that a submanifold isometricaily immersed in IR" has constant mean curvature iff its Gauss map is harmonic. Since t; is minimally embedded in S , it has constant mean curvature in IR , so y is a harmonic map. The composition rz °h is totally geodesic, so the composition map y 'i , -*S is harmonic. In summar,y/or every closed orientable surface Af of genus M — p, the homotopy class in [M, S] of degree 1 — p can be rendered harmonic via the above procedure involving Gauss maps. Of course, we could have started with the polar map d>- :Af->S of the immersion 0:M-*S , described in [25, Sect. 10] as follows: Set d> =d>, and let ±

±

mk

i

4

±

±

n k

3

3

+

110 Harmonic and Holomorphic Maps Belween Surfaces

45

a e T ( M ) be the unit 2-vector of the orientation of T [M): then define 0_(x)eS_ by 0_(x) = *(0(x)Aa(T^ (jW)). If tp is minimal, then d>_ is a harmonic map with a finite number of branch points. Let us note that if we treat 0 <x)eIR , then y(x) = 0 ( x ) A _(x}eA. Thus the bipolar surface [25, Sect. 11] is just our Gauss map y:M-*Gl cA. Lawson has aiso constructed certain closed nonorientable surfaces n with Euler characteristic^-mfc, minimally immersed in S ; here m,k are positive integers with k odd. Again, these have constant mean curvature in IR , so their Gauss maps yri _ -^G (the Grassmannian of planes through the origin in IR ) are harmonic. G , is a nonorientable 4-manifold with G° as its 2-leaved orientable cover. However, G is not a product; furthermore, the above Gauss maps do not lift to G% . In terms of the identification h, we have Wl)

Hxy

U)

4

±

+

2

m k

3

4

4

m

k

i

2

4

2

a 2

tl

^ 4 , 2 ~ (S+ xS_)/(i),

where

i(a , «_) = ( — a , a _ ) . +

+

Remark. For a simply connected complete minimally immersed surface M in IR which is not a hyperplane, Chern [11] has shown that the maps y :M-*S are — holomorphic and have dense images. ±

7. Surfaces of Constant Mean Curvature in IR

4

±

3

(A) Gauss Maps of Branched Immersions 3

{7.1} Proposition. Let M be a Riemann surface, and 0 : M - - I R a branched immersion. Then d> has constant mean curvature iff its Gauss map y:Af-»S is harmonic. That is established in general in [36] for nonsingular immersions. In the present case it is an elementary matter to verify that the Gauss map y is continuous on all M, and (2.6) and (2.10) of [23] imply that it is C and hence smooth; indeed, dy the seond fundamental form of
(7.2) Harm(M,S)/Iso(S) denote the set of equivalence classes of harmonic maps M-*S, two % y being equivalent iff there is an orientation-preserving isometry a of S such that y — a"y. Similarly, let 3

(7.3) Brim^M.^l/IsolIR ) denote the set of equivalence classes of branched immersions with constant mean curvature H. Thus we have the canonical map (7.4)

3

3

Brim^M, R )/Iso(IR )-» Harm(M, S)/Iso(S).

It follows from [1] that this is injective, provided tf#0; that restriction is well known to be necessary, for the catenoid and the helicoid have the same Gauss map. Let us also observe that by Sampson's unique continuation theorem [38], if there is an open UcM and ceSO(3) such that =a

y\v ''y\v< then y =ff->yon all M.

Ill 46

J . Eells and L. Lemaire

Suppose now that M is complete and simply connected; and H is a positive number. Then with every harmonic map y\M-*S there is an explicitly given branched immersion

fR with mean curvature H and Gauss map y, given by Kenmotsu [23]; we are indebted to Banchoff for calling our attention to that work. Indeed, if we let y also denote the map M'->IR defined as y composed with stereographic projection, where M' = A f - y ( 0 , 0 , +1), then 3

2

_1

tb(z) = 2Re Ay o s

+ c,

z

where 2

A=

3

(A\A ,A ) 1

2

. -(1-y )

i(l + y )

-2y

nti+wv

\m+MT

H(i+m

We note that we have the conformality condition i

2

1

1

1

(A ) +(A ) -r(A )

2

= Q;

and 1

2

2

2

3

2

2

2

2

K | + | ^ | + | ^ | = 2//7 (l + | y | ) ; and furthermore, d> satisfies the mean curvature equation ,,_/4Rey

4Imy

U+Irr"

2

2(\y\ -l)\ i+lvl

2

/

on all M. In particular, (7.6) Proposition [23]. For any Riemann surface M, H>0, and harmonic map y.M-rS, we have a canonically defined branched immersion d>:M-*IR with constant mean curvature H. 3

Here M denotes the universal cover of M. 3

Example. If
1

3

WM ^IR . r

These are the associated surfaces of <j>. Presumably this is just the case of y associated with d>.

±

:M-*S

(B) The Gauss Map of Delaunay's Nodoid

In his Thesis [40] Smith constructed analytically certain harmonic maps from certain tori T-rS; these have degree 0, although some of them are surjective. In fact, suppose that T is a flat rectangular torus, expressed as R/flZ x R / 2 J I Z ;

±

112 Harmonic and Holomorphic Maps Between Surfaces

and (S,/i ) the Euclidean sphere of radius I. Let (x,y)e[0,a] x [0,2;r] be Euclidean coordinates on T, and (r,8) polar coordinates on S. A map d>:T--S of the form r =
(7.7)

1

dx

= sin $ cos*.

For a>2n an oscillatory movement of the pendulum of period a induces a harmonic map of degree 0 covering a band around the equator of S. For a suitable choice of coordinates the lines x = 0 and x = a/2 are fold lines of the map. For any value of a, a revolutive movement of the pendulum of period a/2 induces a twice surjective harmonic map of degree 0 from T to S, sending the collapse lines x = 0 and x = a/2 on the poles of S. Calabi has indicated to us that Smith's maps can be interpreted as the Gauss maps of the unduloid and the nodoid of Delaunay [13], see [16, Sect. 11.7] and [44]. The Klein bottle K has the representation as a connected sum of two projective planes; or as a cylinder with antipodal points of its boundary circles identified. Figure 7 is meant to indicate how this representation is related to the usual one:

Fig. 7

A map of the cylinder to S of the form r = $(x), 8 = y subject to the conditions
Using the method of Sect. 4, we see that a nonorientable surface M of even genus can be realised as a 3-sheeted branched covering of K. A suitable choice of conformal structure on M makes the projection conformal; and composition with the hannonic map K—S just constructed produces another harmonic map of degree 1 mod 2 from a surface of even genus to S. To obtain the explicit representation of the harmonic maps K—P announced in Sect. 5Dii), where P is given its Euclidean metric, we choose again Euclidean coordinates (x,y) on K and polar coordinates (r,6j on S:

113 4N

J. Eells and L. Lemaire

Ji X

Fig. 8

a A map tp from the rectangle of Fig, 8 to S of the form dt(x, y) = (4>(x), y) is harmonic iff * satisfies (7.7). A solution exists satisfying the conditions *(0) = 0, e[0,jr], the image of the rectangle covers once the hemisphere. From the identifications indicated in Fig. 8, we check that

8. On Harmonic Maps of Surfaces into Various Homogeneous Spaces Consider harmonic maps (b of S or P into the Euclidean n-sphere S", with n £ 3. In that case tp is a minimal branched immersion by (2.7). 3

(8.1) Every harmonic map tp.S—S factors through a totally geodesic map of an equator into S ; thus their classification reduces to that given in Sect. 3A. See Almgren [2]. In general, we have Calabi's theorem [9, 10; see also 3] that if S—*S ~ '(r)C R" is a minimal branched immersion and if the image lies in no hyperpiane o/R". then n is odd. 3

n

3

(8.2) Every harmonic map S" for n 2:4, A simple example of a harmonic isometric embedding ofP in S* is given by Veronese's map: If (.v,,.Vj,x ) are coordinates in R and (y, y) coordinates in IR , then 3

3

s

5

2

i

immerses S (\j 3) isometrically into S ; and factors to produce ihe desired minimal embedding of P into S . Since every map tp-S--P" lifts to one into S", their consideration reduces to that of the previous case. 4

(8.3) We have Calabi's and Barbosa's theorem [3] '.Assigning the direct ix i^to a map P (or equivalently $-*S ) gives a bijective correspondence between the 2li

2k

114 Harmonic and Holomorphic Maps Between Surfaces

49

harmonic maps
lk

2

(8.4) Every harmonic map P geodesic embedding of P in P

3

is either constant

or factors

through a totally

3

Indeed, lift to S %

S

3

i I p ^ p • 3

now 4> factors, and hence so does
0*0. The maps [25, Theorem 3] (for any positive integers r, s) <£„(*,, x ) = (cosrxj cosxj, sinrx, cosx , cossxjsinxj, sinsx sinx ) are minimal isometric immersions of T=IR/27t2 x R/2JTZ into S . The m a p ^ , defines the Clifford torus T - » S ; (1,1) is the only pair giving an embedding. The map
2

1

2

3

3

s

3

In general [22], ifS" is an isometric minimal immersion of aflat torus whose image lies in no hyperpiane o / R " , then n is odd; furthermore, the image
We obtain minimal immersions of T and K into P" by composition. (8.6) On the other hand, we recall a theorem of Frankel [ 1 8 ] : The inclusion map i of a minimal embedding of a compact hypersurface induces an epimorphism i :jtj(Af)-»jij(W);

M into a space N with Ricci

N>0

#

Therefore (8.7) Any minimal embedding never lifts into S .

into P

3

is homotopically

non trivial;

in particular, it

3

By Corollary (2.7) we obtain: (8.8) The homotopy classes of maps [ M , P"] = H {M; Z )f representatives, since n (P")—0 for n £ 3 . L

2

or

"=

3

a

l

1

have

harmonic

2

As specific examples, Lawson [25] has shown that every orientable M can be minimally embedded in P"(n^3), provided genus M > 0 . Furthermore, every minimal immersion of a compact surface into P" represents a nontrivial homotopy class. (8.9)

Suppose genus M S I ;

then there is no stable minimal immersion of M into S or 3

P . That is a consequence of a theorem of Schoen and Yau [39], which asserts the 3

115

50

J. Eells and L . Lemaire

same conclusion for range any compact oriented 3-manifold with positive scalar curvature. Before considering examples of harmonic maps M->W, let us note the r+ 1

+ 1

(8.10) Lemma. The natural embedding IR C C " given by (x , ...,x,)-<-(x ,..., x,, 0,..., 0) induces a totally real, totally geodesic embedding n: P -*IP" .furthermore n is homotopically nontrivial. 0

0

r

A real vector subspace W of a complex vector space (V,J) is totally real if J{ W)r\ W—0. A real submanifold of a complex manifold is said to be totally real if its tangent spaces are all totally real. Proof. The first assertions are immediate. To prove the final statement it suffices to consider the case*; ;P-»IP . Note that the tangent vector bundle of IP restricted toP decomposes: 2

2

r ( P % = T(P)e-/7'fP). Then n ^ T n ^ n - ' m ^(TW

2

, ) + wJTiPVvw^jnPV

+ WziJTiP))

2

= w (T(P)) = w (r
2

where 1 denotes the generator of H (P, Z). n

(8.11) Proposition. The composition n°d>:M-*P harmonic. If n = 2,

with any harmonic map
deg (0) = deg b7°
2

(8.12) There are many nonconstant harmonic maps S->rP". They are all minimal branched immersions; they are not ail ± holomorphic, but are necessarily strongly pseudoholomorphic, in the sense of Wood. Holomorphic examples are provided by rational functions. Note also that for each n2:2 the composition s i r P is a nonconstant harmonic map of degree 0; in particular, it is not + holomorphic. (8.13) For 2 we have [P, P"] = Z . Lemma (8.10) shows that the nontrivial class is harmonically represented by ff :P-*P ->IP". 2

2

l

l

3

(8.14) The natural embedding S {l/]/3) x S\\l]/l) X S {lfl/3>) in C is a minimal 3-torus in the unit sphere S . The natural action of C(l) on S induces a minimal embedding of a flat torus in P , which is totally real, but not totally geodesic [30]: 5

s

2

S ( V J/3) x S*(l/|/3) x S\XI]/% J

4 r

-» S

5

1" 4

2

• IP .

116

Harmonic and Holomorphic Maps Between Surfaces

51

The left-hand vertical arrow represents a trivial [/(l)-bundle, so 0 is a null homotopic map.

Acknowledgements. During the preparation of this paper the authors enjoyed the hospitality of the Institut des Hautes Etudes Scientiftques. The first-named (J.E.I thanks the Universite de Paris VI, where he wasProfesseur d'Echange. The secon d-named (L.L.) is Charge de recherches au Fonds National Beige de la Recherche Scientifique.

References t. Abe, K., Erbacher, J . : Isometric immersions with the same Gauss map. Math. Ann. 215, 197-201 (1975) 2. Almgren, F.J.: Some interior regularity theorems for minimal surfaces and an extension of Bernstein's theorem. Ann. of Math. 84, 277-292 (1966) 3. Barbosa, J . : O n minimal immersions of S in S Trans. Am. Math. Soc. 210, 75-106 (1975) 4. Barcus, W.D., Barratt, M.G. . O n the homotopy classification of the extensions of a fixed map. Trans. Am. Math. Soc. 88, 57-74 (1953) 5. Berger, M.: Lectures on geodesies in Riemannian geometry. Lecture Notes No. 33, Bombay: Tata Institute 1965 6. Berstein, I , Edmonds, A.L.: Classification of surface maps modulo self-equivalences. Am. Math. Soc Notices '78T-G42 (1978) 7. Blaschke. W.: Sulla geometria differenziaie delle superftcie S nello spazio euclideo S . Ann. Mat. Pura Appl. 28, 205-209 (1949) 8. Brouwer, L . E . J , : Aufzahlung der Abbildungsklassen endlichfach zusammenhangender Flachen. Math. Ann. 82, 280-286 (1921) 9. Calabi, E . : Minimal immersions of surfaces in Euclidean spheres. J. Differential Geometry 1,111¬ 125 (1967] 10. Calabi, E , : Quelques applications de 1'analyse complexe aux surfaces d'aire minima. In: Topics in Complex Manifolds, pp. 59-81 Montreal: University Press 1967 11. Chern, S.S.; Minimal surfaces in an Euclidean space of N dimensions. Diff. and Comb. Top, A symposium in honor of Marston Morse, pp. 187-198. Princeton: Princeton University Press 1965 12. Chern, S.S., Spanier, E . H . : A theorem on orientable surfaces in four-dimensional space. Comm. Math. Helv. 25, 205-209 (1951) 13. Delaunay, G ; Sur la surface de revolution dont la courbure moyenne est constante. With a note appended by M. Sturm. J. Math. Pures Appl. Ser 1, 309-320 (1841) 14. Edmonds, A . L . : Deformations of maps to branched coverings in dimension two. Ann. of Math. 110, 113-125 (1979) 15. Eells, J., Sampson, J.H.: Harmonic mappings of Riemannian manifolds. Am. J . Math. 86, 109-160 (1964) 16. Eells, J., Lemaire, L . : A report on harmonic maps. Bull. London Math. Soc. 10, 1-68, (1978) 17. Eells, J., Wood, J.C.: Restrictions on harmonic maps of surfaces. Topology 15, 263-266 (1976) 18. Frankel, T.: On the fundamental group of a compact minimal submanifold. Ann. of Math. 83,68-73 (1966) 19. Gulliver, R.D., Osserman, R., Royden, H . L . : A theory of branched immersions of surfaces. Am. J. Math. 95 750-812(1973) 20. Heins, M.: Interior mapping of an orientable surface into S . Proc. Am. Math. Soc. 2,951-952 (1951) 21. Hopf, H.: Beitragezur Klassifizierung der Flachen abbil dun gen. J. Reine Angew. Math. 165,225-236 (1931) 22 Kenmotsu. K - On minimal immersions of S into S". J . Math. Soc. Japan 28, 182-191 (1976) 23. Kenmotsu, K . : Weierstrass formula for surfaces of prescribed mean curvature. Math. Ann. 245. 89¬ 99 (1979) 24. Kobayashi, S., Nomizu, K . : Foundations of differential geometry. I. I i . New York: tnterscience 1963, 1969 2

2m

2

1

2

4

117 52

25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44.

J. Eells and L . Lemaire

Lawson, H.B.: Complete minimal surfaces in S?. Ann. of Math. 92. 335-374 (1970) Lemaire. L . : Applications harmoniques de surfaces. C.R. Acad, Sci. Paris A280, 897-899 (1975) Lemaire, L . : On the existence of harmonic maps. Thesis, Warwick University 1977 Lemaire, L . : Applications harmoniques de surfaces riemanniennes. J. Differential Geometry 13, 51-78 (1978) Lemaire, L . : Harmonic nonholomorphic maps from a surface to a sphere. Proc. Am. Math. Soc. 71, 299-304 (1978) Ludden, G.D., Okumura, M., Yano, K . : A totally real surface in CP that is not totally geodesic. Proc, Am. Math. Soc. 53. 186-190 (1975) Morrey, C.B.: The problem of Plateau on a Riemannian manifold. Ann. of Math. 49, 807-851 (1948) Morrey, C.B.: Multiple integrals in the calculus of variations. Grundlehren Band 130. Berlin, Heidelberg, New York: Springer 1966 Olum. P.: On mappings into spaces in which homotopy groups vanish. Ann. or Math. 57, 561-574 (1953) Olum, P.: Mappings of manifolds and the notion of degree, Ann. of Math. 58, 458^180 (1953) Olum, P.: Cocycle formulas for homotopy classification; maps into projective and lens spaces. Trans. Am. Math. Soc. 103, 30-44 (1962) Ruh, E.A., Vilms, 3.: The tension field of the Gauss map. Trans. Am. Math. Soc. 149,569-573 (1970) Sacks, J., Uhlenbeck, K . : The existence of minimal immersions of the two-sphere. Bull. Am. Math. Soc. 83, 1033-1036 (1977) Sampson, J.H.: Some properties and applications of harmonic mappings. Ann. Ecole Norm. Sup. XI, 211-228(1978) Schoen, R., Yau, S.T.: Existence of incompressible minimal surfaces and the topology of three manifolds with non-negative scalar curvature. Ann. of Math. 110. 127-142 (1979) Smith, R.T.: Harmonic mappings of spheres. Thesis, Warwick University 1972 Tomi, F . : Eineinfacher Beweiseines Regularitatssatzes fur schwache Losungen gewisser elliptlscher Systeme. Math. Z. 112, 214-21B (1969) Wood, J . C . : Singularities of harmonic maps and applications of the Gauss-Bonnet formula. Am. J. Math. 99. 1329-1344 (1977) Wood, J.C.: Holomorptncity of certain harmonic maps from a surface to complex projective nspace. J. London Math. Soc. 20, 137-142 (1979) Eells, J . : On the surfaces of Delaunay and their Gauss maps. IV. Coloq. Inter, Geom. Dif.(in honour of E , Vidal), Santiago de Compostella, pp. 97-116(1979)

Received April 15, 1980

2

118

Proc. Indian A c i d . Sci. (Math. Sci.), Vol. 90, Number i , February 1981, pp. 33-45. ©

Printed in India.

Deformations of metrics and associated harmonic maps By J E E L L S and L L E M A I R E

1, Introduction In this paper wo establish a theorem ((3.1) below) concerning the dependence of harmonic maps on the Riemannian metrics used to define them. Our method is an application of tbe implicit function theorem in suitable manifolds of maps, In the case of a compact range of negative curvature, that result was obtained by Sampson [24], using derivative estimates for harmonic maps given in [10], Another special case has been found by Schoen-Yau [26], and applied by them to obtain theorem (6.3) below. Their proof is entirely different; it is based on a Dirichlet growth theorem of Morrey ([21] lemma 9.4.18) and thus is restricted to maps with two-dimensional domains. In some cases the geometry of the problem introduces irrelevant degeneracy. For instance, that can happen if M or JV" has a nontrivial group of isometries— or if the harmonic map under consideration factorises through a geodesic. More technical results concerning some of these cases are given in § 5. As applications, we offer proofs of two theorems on harmonic diffcomorphisms of surfaces, due to Sampson [24] and Schoen-Yau [26]. We are much indebted to Professor Sampson for many stimulating conversations, as well as for making available to us (many years ago) a preliminary version of his manuscript [24]— a primary source of motivation for the present work.

2.

Basic properties

(2.1) L e i M and TV be smooth (i.e. C") finite dimensional manifolds, with M compact. Unless otherwise specified, we suppose M and N without boundary. We denote by T(M) and T* (Af) the tangent and cotangent vector bundles of M. We are primarily interested in smooth Riemannian metrics on M and N and their associated smooth harmonic maps. However, our main tool is the implicit During the preparation of this paper the authors enjoyed the hospitality of the Institut des Bautes Etudes Scientifiques. The first-named thanks the Universite de Paris V I , where he was Professeur d'Echange. The second-named is Charge de recherches au FoDds National Beige de fa Recherche Scicntifique.

33 P. ( A ) - 3

119 34

J Eells and L Lemaire

function theorem, which in its simplest form applies to maps between Banach spaces. Therefore, at times we shall restrict ourselves to finite orders of differentiability. Furthermore, we need to solve certain linear elliptic equations, and we choose to do so in Schauder's context; see for instance [12]. Therefore, for r an integer such that 2 ^ r <, oo and 0 < a < 1, we shall consider the space C ° (Af, /V) of maps whose differentials of orders <, r satisly an -H61der condition. With the associated topology, C (Af, N) is a smooth manifold modelled on a Banach space, provided r < co. Let lVr • and N " be the spaces of C'-^-Riemannian metrics on M and N. These are open subseU of the Frechet spaces C ( O T* (M)) and C * ( O T* (A/)). If r < DO, M « is an open subset of a Banach space, since M is compact. +

Q

+

+

a

r +

+

+

8

a

2

r+

+,+

, + , +

(2.2) Given (g,A) e M ' » x N « and a C'+^-map tp: M -» N(sometimes written tp : (A/, g) -# (N, ft)), its energy is denned by £ ( p ) = £ ( > ; g , A ) = i J \d
Then

E :C

+ 3 + a

( A / , TV) - * ( > 0) is a C'+^Munction.

(2.3) The extremals of E are called harmonic maps, and satisfy the Euler-Lagrange equation [10] Tp - 0; i.e.

V , £(¥>)= -

S (r
= 0

r + S +

+ a

for all variations v e T ( C * (A/, TV)) of y . T is an element of ( C ' (Af, N)) called the tension field of p. At times we shall have to specify the metrics used and write T ^ = T iff 8> ft)- 1° I l coordinate systems {x } and {«°} on Af and Af, is represented by v

9

Q c a

1

where * T and T are the Christoffel symbols of g and ft. Given a harmonic map p : M -*. N, the second variation of its energy is described by the Hessian y*E(tp), characterised by the identity [20], [29] V * £ ( p ) (u, * ) = / < JfV(x% • where —

is the ./at-ofti operator along

w(x))dx, represented in the notations of ([9], §3

by - J (v) = Trace 9

v + Trace R" {df, v) dp,

where Rf is tbe curvature tensor of (N, ft), /p is a linear elliptic operator. Integration by parts, using the symmetry properties of R", shows that ; {JyV, w) dx= that Jq> is self-adjoint.

J {v J w)dx; t

9

i.e.,

120

Deformations of metrics and harmonic maps

35

2

The condition that S? E(p) is non-degenerate is that K e r / ^ ^ O (or equivalently Ker Jy = 0). The first equality implies that tp is an isolated critical point of E and the second that +e

+

J

) implies therefore that is an isomorphism. (2.4) For our application of the implicit function theorem, we consider the tension field r defined in (2.3) as a map 9

T:C

r + a + a

1+

(M, N) X M'+ » X N ' +

t + l +

° -» T(C'

4a

(M, N)).

We work in a neighbourhood of a C'-harmonic map ip„ : (M, g ) -»(N, A ) where g and h are C, but in order to remain in the context of Banach spaces, we take r < co and restrict attention to an exponential chart U of C (Af, N) centred at p and such that the image of every map in the chart is contained in a fixed compact domain K of N. By the canonical identification through exp^ (using the metric ft ), we have 0

a

0

B

r + a - i o

a

0

T(C'+« {M, AO) |

u

= U x r

f t

( C ' « (A/, Af)).

We can represent r by its composition with the projection on the second factor, which we still denote by t : r+1

f

+1+

(2.5) * : U x M + ° X N'- * " -+ T

(C

n

+ a

(M, AO).

T is now a C*-map from an open set of a Banach space to a Banach space. (2.6) Lemma. Consider a smooth path {
v

fe&fti

*

M

'

+ w

r +

* x N *

starling at the given C*-triple (y> ;g ,h ) 0

0

0

+ 1 + 0

U

and put

X

~Js-

Ds

i

and

Z= " *

ivAere T, and T, are the Christoffel symbols associated to g, and A,. (Y depends on X, and is given by n,= //

| g*» ( 7 .

+ V , X» ~ 7 , *«,)).

p : (A/, g ) -*• (N, h„) a

(2.7)

is harmonic, then

0

d-r I ^

Trace, (y

s

= dr dpi; g , A ) ( r , * , Z) == Aw + Trace R" (d
0

a

pi) - Trace (Ydtp ) + Trace Z C# a

M

d^ ). a

The terms depending on « are derived in [29], and the others in [19].

121 36

/ Eells and L Lemaire

3.

The main dependence theorem

(3.1) Theorem : Suppose that p : (A/, g„) - • (TV, h ) is a C-harmonic map between C"-Riemannian manifolds such that S7'E(p ) is non-degenerate. Then for 1 < k < oo and r < oo there is a neighbourhood V of (g , A ) in M'" * *" X N +*+ + and a unique C-map S : V -* C (A/, TV) such that S (g„, A ) = j>, S (g, A) is a harmonic map tp ! (A/, g) -* (TV, ft). From (2.7) we note that the hypothesis on the Hessian is satisfied if the sectional curvature Riem" is strictly negative and p (A/) is not a point or a geodesic. 0

a

t

1

0

r

I

a

1

0

+ B + a

0

0

(3-2) Corollary: Lethal and r < oo. Consider a C-path (g„ A,)„ , of C-.mlrics in M'* " x N ' * and suppose that for each sel there is a unique harmonic map tp, : (Af, g ) -» (JV, A,) i« a fixed homotopy class, with each y' E(tp,) non-degenerate. Then the map I - J - C" " * (A/, AO given Aj> s -*
1+

+

w + a

1

(

1

2

0

r

(3.3)

s

Remark : In particular, the conclusion of (3.2) is valid if

(a) each ft, has negative sectional curvature; (b) each (TV,ft.)satisfies the growth conditions described in ([9]; (5-2'a))— always satisfied if TV is compact; (c) the homotopy class contains neither constant maps nor maps to closed geodesies. Application : Such continuous dependence in dimension 2 has been used by Earle-Eells (6) to construct a specific continuous section of the fibre bundle of complex structures over the Teichmiiller space of a closed surface. Proof of theorem (3.1). Consider the map T defined as in (2-5). From (2.7) and the non-degeneracy of v E( deduce that its partial differential in the U direction is an isomorphism. By the implicit function theorem in Banach spaces [5] there is a neighbourhood V of (g ,A») in M ' " X \K and a unique C*-map S : V -* C (A/, AO such that S (g„, A ) = q> and t (S (g, A) ; g, A) = 0 for all (g,ft)£ V. The corollary follows easily by covering the path with neighbourhoods V constructed as in the theorem and using the unicity of the harmonic maps

w

e

+ i +

0

r + 2 + a

0

Q

t

4

U

(3.4) Remark. Real analytic ( = C ) harmonic maps have properties not generally shared by other C'-maps [32]. Let Af and TV be C"-manifoIds. It is known ([31], [23]) that the C-metrtcs are dense in M * x N". Furthermore, if tp : (A/, g) -* (TV, A) is a harmonic map between C -Riemannian manifolds, then tp e C°. Theorem (3-1) shows that any C™-harmonic map f between C -manifolds w

u

122

Deformations of metrics and harmonic maps

37

with C°°-Riemannian metrics for which y * £ ( p ) is non-degenerate can be Capproximated by C-maps which are harmonic with respect to suitable metrics.

4.

Dependence for the Dirichlet problem

(4.1) Suppose now that M has a boundary I M and consider the Dirichlet problem (4.2) if

=0 tp

|

=y>

DM

for a fixed map i c e E " (DM, TV), tbf space or restrictions to DM of maps in C - (M, TV). We denote by C$ (U, TV) the Banach manifold of maps tp . M -* N coinciding with \p on DM, r < oo. Its tangent bundle A

T(Cf(M,N))

={veT(C'+*(M,N))

!« \ba = 0 and

nov\ =y,}, QB

where JT : T'TV) -* TV is the canonical bundle projection. The Jacobi operator dip is now a map 2+a

Ttp (C$

(A/, TV)) -» T

For

v,weT
integration by parts as in § 2 shows that J" {Jq> v, w,dx = / (v, Jf w) dx, M H and the non-degeneracy of V E(f) means again that ftp is an isomorphism. 2

(4.3) With these modifications in the proofs, theorem (3.1) and corollary (3.2) remain valid when the harmonic maps are solutions of the Dirichlet problem (4-2), replacing C ' ^ ( M , N )

hI+

by C5 °(A/,7V).

Furthermore, remark (3-3) remains valid for the Dirichlet problem without conditions (6) and (c). Indeed, the existence and unicity of the solution to that problem are guaranteed simply by the curvature assumption (a) ([25]; the unicity follows also from [15]). (4-4) In [13] Goldstein establishes the smooth dependence cn tp of the solution of (4-2) in a neighbourhood of a hannonic map, under the hypothesis that Riem <, 0. His method uses the inverse function theorem and applies to ihe case of a harmonic map with non-degenerate Hessian (without hypothesis on the curvature). Adapting his proof, we see that in a neighbourhood in C ' " (M, AO of a map ¥»„ with V E ( f ) non-degenerate, the map

w

+ w

!

t

A : M'

+ 1 + a

r+i+:<

M

x N'+ x

m + a

f^m-m

+ 2

x C + ° (M, AO -* x

rp„ C '

+ a

r

S

(AT, TV X E -+ +»(S Af.JV)

defined by A (g, h,
123 38

/ Eells and L Lemaire

tp (Af, g ) -> (TV, /i ) is a C-solution of (4.2) with boundary data ip such that V £ ( $ » o ) is non-degenerate, then there is a neighbourhood V of (g , h„; y ) in M'+ - X iV+*+ +* X C ° (Af, TV) and a C-map V -* (A/, AO JUCA that S(g ,h ; \j/ ) — a}^; and S(g,h,i//) is a solution of (4.2) for all (g,h;\/j) e V . See ([16]; Thm 10.5.3) for an alternative treatment in tbe case N = R. a

0

0

a

3

0

I(

0

a

0

l

0

r + 2 +

e

a

(4.5) Example. A simple instance of degeneracy of V £ ( p ) is the following. Take Af — / , N = S" (— the Euclidean n-sphere, n > l j a n d ^ u carrying the endpoints of I onto two antipodal points of S" In this case the solutions of (4.2) are geodesic segments f. We observe that tp is not a continuous function of \j> at yf . B

9

(4.6) Problem. When is { f e E (aAf, N) : E : CJ° (Af, AO-* * is non-degenerate at every critical point} residual (i.e., a countable intersection of open dense subsets) in the space of y for which (4.2) has a solution? See ([22], ch. VII) for the case M = /. (4.7) Problem. In the same order of ideas, when is {(g, I I ) G M ° X N " : the function tp -* E(tp;g,h) in non-degenerate} residual? In that problem one may have to discard certain degeneracies, as was done by Abraham in [2]. (4.8) Example : Let tp : M -» N be a harmonic map. A theorem of Visik [30] applied to the operator J«, asserts that there is a number e > 0 such that for any smooth domain A f C Af of volume M' < we have V*#(j> | ifef') non-degenerate. e

5.

Killing degeneracies %

(5.1) Degeneracy of X} E(tp;g,H) is expressed by the presence of non-zero Jacobi fields along
defined by fayr) -*y/o
0

m

a

+2l

M

a

0

0

r

s

Proof. As in (5.2), I(N,h„) acts smoothly on C + +° (Af, A). By the hypothesis of surjectivity, this action is free on the component of

124

Deformations

of metrics and harmonic maps

39

For a chart U centred at f , let V = V/I(N, h ) denote the orbit manifold of /(TV, A ). Because the tension r is invariant under the action of /(TV, A ), it induces a C - m a p a

a

0

0

r

0

r : U X M +'+ -* T-

(Af, N)II(N,

A„))

as in (2.5). Since Ker — I (TV, A ), the partial differential of T at p in the I I direction is an isomorphism. Therefore, there is a neighbourhood V of g in 0

0

tt

+ , 4

M ' ° and a unique C - m a p S :V-*fj such that S(g ) = p and r(S(g%g) = 0 for all g e V. Following S (g) by the action of /(TV, ft ), we obtain the S of the proposition. 0

0

0

(5.4) By way of contrast, we see from (5.2) that I(M,g„) acts continuously but not differentiably on C (Af, A ). Consequently, we are unable to proceed as in (5.3). However, a satisfactory result can be obtained in an important case, found in the classical literature under the heading of Poincare's continuation method (see [22; Cb. IX] and [27; §21]). The case under consideiation is that of a non-trivial closed geodesic represented as a harmonic map p : S -*(N,h). Now the identity component I(S )'=S induces a one-dimensional degeneracy of V B(p), but does not act differentiably on C ° (S , TV). We follow the usual practice in ealling rp non-degenerate if its only Jacobi fields are those of I (S ). Tbe following result is surely known (We express our thanks to W Klingcnberg and D Sullivan for their comments.) j f + 2 + a

7

1

1

J

2

+ 2 +

1

1

(5.5) Proposition. The non-degenerate smoothly on small deformations of h.

closed

geodesies

of

(TV, A)

depend

Indeed, we identify the prime closed geodesies of (TV, A) with the periodic orbits of the geodesic flow on 2" (TV). The non-degenerate closed geodesies correspond then to the periodic orbits whose Poincare maps have no eigenvflue equal to one. One then applies tbe proof of ([27], §21), or of the stability theorem ([3], §24) to obtain the desired conclusion. (5.6) Let p :(A/,g ) "* (TV,A ) be a harmonic map which factorises through the circle S and whose only Jacobi fields are those induced by I (S ). Combining (5.3) and (5.5), we see that the hannonic map will depend smoothly on a sufficiently small smooth deformation of the metrics. The interest of this observation is that, if we exclude the trivial case of constant maps, that kind of degeneracy is the only possible one in the case Riem < 0. In particular, we can delete the condition (c) in remark (3.3). 0

0

0

1

1

w

6.

Harmonic difleomorphisms of surfaces

As an application, we give a new proof of the following result of Sampson [24] and Schoen-Yau [26] (their proofs involve a decomposition formula for the Laplacian of | dtp | : Af R and the maximum principle). 2

125 40

J Eells and L Lemaire

(6.1) Theorem. Let (Af, g) and (TV, A) be closed orientable surfaces of the same genus p^>l. Assume that the sectional curvature Riem" < 0. If tp : Af -* TV is a harmonic map of degree 1, then tp is a diffeomorphism. Proof. Suppose first that p > 2, so that Riem" ^ 0 . Our hypothesis ensures that tp is homotopic to a diffeomorphism tp . Let us consider a smooth path (g,\ , C M " such that g„ = tp*„ h and g = g. The diffeomorphism tp : (Af, g )-» (TV, A) is then harmonic. Since Riem" <, 0 and 0, for all s e I there is a unique harmonic map f, : (Af, g,) -* (TV, A) homotopic to tp„ [10], [15]. By unicity, p is the given diffeomorphism and p = tp. Furthermore, y^fifft) is non-degenerate, so that corollary (3.2) implies that (fp,),ei' continuous deformation of ? to p. Let S = {t : / ( f t ) > 0 for s < t] where / ( » , ) is the Jacobian of tp,. Since a map of degree 1 with positive Jacobian is a diffeomorphism, it suffices to show that S — I. Therefore we remark that Oe S and show that S is open and closed. That S is open follows from the compactness of Af and the continuity of the above deformation. To show that S is closed, we consider a sequence (/,) c S such that ! ( - * * » • Then J(ip )^.0. By Wood's classification of singularities [32], tp must be a branched covering. Since it is of degree 1, it is a diffeomorphism and r„ e S. a

e

t

0

0

0

x

s

a

a

la

lai

The case p = 1 follows from standard methods (see [9], § 7). By passing to the two-sheeted orientable coverings, we get the (6.2) Corollary. Let (M,g) and (TV, A) be non-orientable closed surfaces of the same genus # 2 * 1 . Assume Riem" <, 0. Any harmonic map £ : (Af, g ) - » (TV, A) homotopic to a diffeomorphism is a diffeomorphism. Once in possession of (4.3), we can simplify somewhat the proof of the fallowing result of Schoen-Yau [26] : (6.3) Theorem : Let (Af,g) be a compact surface with boundary and (TV, A) a compact surface with Riem £ 0. Suppose that ip : Af -» TV is a diffeomorphism onto (Af). Assume that y (£iAf) is convex with respect to (f(Af). Then there is a (unique) harmonic map tp : Af -» TV in the component of tp in C$ (Af, TV) such that q> ] H « -*•

0 and J ( f t ) ^ 0 ior s < i}. As above, S is closed; and we observe that the method of Schoen-Yau applies to this deformation to show that S is open : Indeed, use the complex structure of M to decompose J dtp, | = | d' tp, | -f- | d" tp, | . Take tn^S; then the continuous dependence theorem ensures that \d'0 on Af for all t near t„. The proof of theorem 3.1 of [26] now applies, to show that / ( f t [ H ° ) > 0; we conclude that a neighbourhood of f„ is contained in S. w

l O U

e

i

0

2

s

2

2

7.

Hannonic maps with non-negative Jacobian

We now examine briefly the possibility of extending the results of § 6 to other situations.

126 Deformations of metrics and harmonic maps

41

(7.1) It is unknown whether the curvature restriction in theorems (6.1) and (6.3) is necessary. (7.2) Almost nothing is known about the singularities of harmonic maps between manifolds of dimensions > 2. We note, however, that for n ^ 3, there are metrics g on the n-torus such that any harmonic map