The Ricci Flow: Techniques and Applications: Geometric Aspects (Mathematical Surveys and Monographs) (Pt. 1)

f (t) = fo

(1.12)

0

cp(t) =
Moreover, (1.13)

Rc (g (t))

+ V'g(t)V'g(t) f

(t)

+ ;7 g (t) = 0,

where V'g(t) denotes the covariant derivative with respect to 9 (t), and af (t) = 1gradg(t) f (t) 12 . -a t get)

(1.14)

EXERCISE 1.8. Prove Proposition 1.7. HINT: Verify the equations in the order in which they are presented. Because of the proposition above, we shall at some times consider gradient Ricci soliton structures and at other times consider gradient Ricci solitons in canonical form. Now we give a couple of examples of Ricci solitons in canonical form. We are already acquainted with solutions which evolve purely by homothety, e.g., these solutions correspond to Einstein metrics. If go is an Einstein metric with Einstein constant A (Le., Rc(go) = Ago), then

g(t) = (1 - 2At)go satisfies the Ricci flow

a g = -2 Rc(g), m

since Rc(g) = Rc(go) = Ago. An Einstein manifold with A > 0 is necessarily compact (thanks to Myers's theorem) and, as the metric approaches zero, the manifold shrinks to a point in finite time. Einstein metrics are stationary points for the normalized Ricci flow on a closed manifold. Following up on our previous discussion of static Euclidean space, we have the following. EXAMPLE 1.9 (Gaussian soliton). Regard Euclidean space as a Ricci soliton in canonical form, so that (1.4) holds with CT(t) = 1 + ct for some c E R For c '" 0, the Euclidean solution is called the Gaussian soliton. By differentiating (1.4) and multiplying the result by CT(t), we obtain (1.15)

0 = -2 Rc(gcan) = D"fgcan + cgcan,

where (1.16)

f(x)=_c 1x I2 4

1. RICCI SOLITONS

6

is the exponent in the Gaussian function. Here we used the standard identity D"jgcan = 2,1''\1 f. 2. Differentiating the soliton equation analysis

local and global

2.1. Differentiating general solitons. Let (g, X) be a Ricci soliton structure (gradient or not) on Mn: (1.17)

2~j

+ 'ViXj + 'VjXi + Egij

=

O.

Condition (1.17) places a strong condition on 9 and X. For example, contracting (1.17) with 9 (tracing) gives (1.18)

R + div X

+ ~E

= 0,

where div X ~ gij'ViXj and R is the scalar curvature. If M is closed, then this implies 2r E = --,

(1.19)

n where r ~ iM R d{L/ vol(M) denotes the average scalar curvature. Furthermore, we also have the following. LEMMA 1.10. If (g, X) is a Ricci soliton structure on Mn, then (1.20) or more invariantly,

~Xi

+ ~£ lm Xm = 0,

~Xo +Rc (x o) = 0,

where Rc : T* M - T* M is defined by Rc (a)i ~ ~£ lmam . PROOF. Taking the divergence of (1.17) and applying the second contracted Bianchi identity (VI-3.13) and the Ricci identity (Vl-p. 286b), we obtain

£) mXm + ~Xi

= g3'k ( 'Vi'VjXk - Rjik£g

= -'ViR+~£lmxm+~Xi'

where we have used (1.18). The lemma follows from cancelling the -'ViR terms. 0 Lastly, computing the scalar curvature of the evolving metric (1.11) and comparing its time-derivative (at t = 0) with its evolution under the Ricci flow gives LEMMA 1.11. If (g, X) is a Ricci soliton structure on Mn, then (1.21)

~R + 21 Rc 12 = CxR - ER.

2.

DIFFERENTIATING THE SOLITON EQUATION

7

PROOF. The left-hand side (LHS) is just the usual expression for ~~ under the Ricci flow. To obtain the equality, we compute ~~ by way of the equality R(Tl.p*g) = -r- 1
(1.22)

b.R - (\1 R, X)

+ 21 Rc 12 + cR = O.

We rewrite this as (1.23) b. ( R +

~c) - \\1 ( R + ~c) ,X) + 21 Rc + ~ g 12 -

c(R + ~c) = O.

At any point Xo E M such that R (xo) = R min , we have 21Rc +~gI2

_ c(R + ~c)

::; O.

If (g, X) is expanding or steady, then c ~ O. Since R(xo) + ~e = R(xo) - r is nonpositive, this implies IRc +~gI2 (xo) = O. Tracing then implies nc

Rmin

= R(xo) = -2 = r,

and hence R == r = - ~e. Substituting back into (1.23), we conclude that Rc == -~g. If (g, X) is shrinking, then c < O. Applying the weak maximum principle to (1.22), we have R ~ O. By the strong maximum principle, either R == 0 or R > O. If R == 0, then (1.22) would imply Rc == 0, contradicting the assumption that (g, X) is shrinking. Hence R> O. D We will return to the consideration of shrinking solitons on closed manifolds in Section 7 of this chapter.

2.2. Differentiating gradient solitons. The gradient Ricci soliton structure equation (1.24)

relates the Ricci tensor to the Hessian of f. First note that tracing this gives nc (1.25) R + b.f + 2 = o.

1. RICCI SOLITONS

8

Now we proceed to systematically differentiate the function f up to fourth order, commute pairs of covariant derivatives, trace, and apply (1.24) and the second Bianchi identity to relate derivatives of the curvature and f. We begin with considering three derivatives of f since for one derivative there is nothing to do, and a pair of covariant derivatives acting on a function commute. Since ''ih [\7 i , \7 j ] f = 0, the only nontrivial commutator is \7 i \7j\7kf - \7j\7i\7d = [\7 i , \7j ] \7kf.

By (1.24), the commutator formula, and \7g = 0, we have (1.26) Taking the trace by multiplying by gjk and using the contracted second Bianchi identity, we get (1.27) Note that (1.26) is antisymmetric in i and j, and in particular, (1.27) is the only equation obtained by tracing (1.26). Next we consider equations obtained by commuting four derivatives of f. The only essentially new equation is obtained by considering \7 i [\7 j, \7 k] \7 d. The quantity \7 i \7 j [\7 k, \7 e] f is zero and [\7 i, \7 j] \7 k \7 d yields only a standard commutator formula. We have

which implies

and hence (1.28) First we trace by gij. Commuting derivatives and applying the contracted second Bianchi identity yield b.Rke -

1

"2 \7 k \7 eR + 2~kem~m -

E

RkmRme

+ "2 Rke

= \7 iRikem \7 mf = (-\7eRkm + \7 mRke) \7 mf. Since (1.28) is antisymmetric in j and k, we have that tracing (1.28) by gjk is zero, whereas tracing by gik is equivalent to tracing by gij. The remaining trace is by gil. We leave it as an exercise for the reader to show that taking this trace yields (-\7jRkm + \7kRjm) \7 m f = 0, which is nothing new since it follows directly from (1.26).

2.

DIFFERENTIATING THE SOLITON EQUATION

9

In conclusion, the new identities we obtain by differentiating fourth order are (1.29) (1.30)

-\1iRjk

+ \1jRik =

f

up to

-Rijke\1d,

\1i\1jRkf - \1i\1kRje = \1i R jkem \1 mf - Rjkem (Rim

+ ~gim)

and the traces are 1

"2 \1i R =

(1.31 ) (1.32) (1.33)

Rie\1d,

+ \1mRkf) \1mf,

M ke = (-\1f Rkm tlR + 21Rcl 2

+ eR =

\1f· \1R,

where

1 Mkf ~ tlRkf - "2\1k \1e R

+ 2RikfmRim -

RkmRmf

e

+ "2Rkf'

The last equation, which is the trace of the equation above it, is a special case of (1.21). REMARK 1.14. The quantities in (1.29) and (1.32) appear in the matrix Harnack quadratic discussed in Part II of this volume (see also subsection 4.2 of Appendix A). We will finish this subsection with an application of Lemma 1.10 to gradient solitons. Substituting X = \1 f in (1.20) yields tl(\1d)

+ Rifg fm \1 mf = o.

On the other hand, commuting covariant derivatives gives tl(\1d) = \1i(tlf)

+ Riklf\1ff.

Combining these equations with (1.25) and (1.9) yields 0= \1iR + 2gkf\1 i \1 kf\1d

+ e\1d =

\1 i (R + 1\1 fl2

+ ef),

proving the following. PROPOSITION 1.15 (Constant gradient quantity on solitons). 1f(g, \1 f, e) is a gradient Ricci soliton structure on a manifold Mn, then (1.34)

R + 1\1 fl2

+ ef == const

is constant in space. Consequently, by (1.25), (1.35)

R + 2tlf - 1\1 fl2 - ef

== const.

Formula (1.34) is used in the study of the geometric properties of gradient Ricci solitons; see Chapter 9 of [111] for an exposition. A significance of the quantity on the LHS of (1.35) will be exhibited in the use of (5.42) and (5.43) to prove energy monotonicity in Chapter 5. With additional hypotheses, we can determine the constant in the proposition above (see also Theorem 20.1 in [186]).

10

1. RICCI SOLITONS

COROLLARY 1.16. If (g, V J) is a steady gradient Ricci soliton structure on Mn with positive Ricci curvature and if the scalar curvature attains its maximum at a point 0, then

R+

IVfl 2 = R(O).

PROOF. We have Rij = -ViVjf > 0 and by Proposition 1.15, R+IV fl2 is constant. On the other hand, at 0 we have 0 = Vi IV fl 2 = -2RijgJ'k Vkf, which implies that V f (0) = O. 0 REMARK 1.17. If we only assume the Ricci curvature is nonnegative, then there is a counterexample by simply taking the flat metric on the xyplane and letting f be the x-coordinate, in which case Rij = - Vi V j f = 0, so that R + IV fl2 = const > R(O) = O. Quantities which are constant on gradient Ricci solitons are useful in the study of their geometries; see Chapter 9 of [111] for an exposition of some examples of this. 2.3. Ricci soliton structures and exterior differential systems. We end this section with a local analysis of Ricci soliton structures. From (1.17), which defines a Ricci soliton structure (g, X), one might hope to obtain more stringent tensoriallocal conditions on 9 and/or X by a combination of differentiation, contracting, and equating mixed partials. However, no further lower-order (Le., first-order in X and second-order in g) identities arise this way. This follows from applying the machinery of exterior differential systems to the soliton equation (1.17), written in local coordinates as a system of PDE that is second-order in the entries of 9 and first-order in the components of X. The theory of exterior differential systems - in particular, the Cartan-K8.hler Theorem3 - is able to predict when a system of PDE has solutions, and how large the solution space is, provided the system passes a test indicating that it is involutive. 4 In addition, if the system is involutive, then any k-jet of a solution can be extended to a (k + I)-jet of a solution, and so on to a convergent power series solution. In the case of (1.17), in order to obtain an involutive system, one has to reduce the enormous size of the solution space (which is due to the diffeomorphism invariance of the soliton condition) by adding the requirement that the local coordinates are harmonic functions with respect to the metric g. Since ~ = gij (OiOj - rfj Ok) , this is a first-order condition on the entries of 9 and takes the form (1.36)

gijrfj = 0,

where rfj are the Christoffel symbols. This is just another variation on DeTurck's trick for proving short-time existence for the Ricci flow (see 3See Theorem III.2.2 and Corollary II1.2.3 on pp. 80-88 in [36] or Theorem 7.3.3 on pp. 254-256 in [223]. 4See Chapter III, pp. 58-65 of [36] or Chapter 7, p. 256 in [223].

3. WARPED PRODUCTS AND 2-DIMENSIONAL SOLITONS

11

Sections 4.3 and 5 of Chapter 3 of [108]). Once this condition is added to (1.17), the system becomes involutive (see [221] for the proof). Furthermore, involutivity implies that Cauchy problems for the augmented system (1.17) and (1.36) are locally solvable; for example, the I-jet of 9 may be arbitrarily prescribed along a hypersurface transverse to a given vector field X. Thus, Ricci soliton structures (g, X) locally depend, modulo diffeomorphisms, on n(n + 1) functions of n - 1 variables (i.e., the components of g, and their transverse derivatives, along the hypersurface). 3. Warped products and 2-dimensional solitons

In this section, we review some of the examples of 2-dimensional Ricci solitons, such as the cigar, which have been constructed to date; the Bryant soliton is discussed in Section 4 of this chapter and examples of Kahler-Ricci solitons will be discussed in the next chapter. In Section 1 of Appendix B we will give some conditions under which the Ricci flow converges to some of the solitons discussed here. Recall from Lemma 5.96 on p. 168 of Volume One that any ancient solution of the Ricci flow on a surface of positive curvature that attains its maximum curvature in space and time is the cigar. Similarly, any ancient solution of the Ricci flow on a manifold of positive curvature operator that attains its maximum curvature in space and time is a steady gradient Ricci soliton. In dimension 3, it is conjectured that such a soliton is the Bryant soliton. This is one reason for focusing our attention on the cigar and Bryant solitons. 3.1. Solitons and Killing vector fields on surfaces. If (g, X) is a soliton structure on a surface M2, then X is a conformal vector field. This is simply because ViXj + VjXi = - (R + c:) gij' If (g, 'V f) is a gradient soliton, then J ('V f) is a Killing vector field (where J : T M ~ T M is the complex structure, defined as counterclockwise rotation of tangent vectors by 90°). To see this, we observe that since 'V f is a conformal vector field on a surface,

(CJ('\lf)g)ij

= Vi (Jj'Vkf) + 'Vj (JikVkf) = Jj'Vi'Vkf + Jik'Vj'Vkf = - (

~ + ~) (Jj gik + Jik9jk) =

0,

where the components J! are defined by J (8~i) ~ J! 8~j' That is, J ('V f) is a Killing vector field. In dimension 2 we have the following. LEMMA 1.18. A surface with a Killing vector field is locally a warped product. In particular, a gradient soliton on a surface is locally a warped product.

12

1.

RICCI SOLITONS

PROOF. Let (M2, g) be a Riemannian surface (not necessarily complete) with a Killing vector field K. Let x E M and define a smooth unit speed path'Y: (ro - E, ro + E) -+ M by 'Y (r) = I~~~~I ("( (r)) , 'Y (ro) = x. Define a I-parameter family of smooth paths !3r : (00 - E, 00 + E) -+ M by /3r (0) = K (!3r (0)), !3r (00 ) = 'Y (r), and a I-parameter family of smooth unit speed paths 'Yo : (ro - E, ro + E) -+ M by 'Yo (r) = I~~~~I ("(0 (r)), 'Yo (ro) = !3ro (0). Note that 'Yoo = 'Y. CLAIM. !3r (0) = 'Yo (r). PROOF OF CLAIM. This follows from

J (K) ] [ K, IJ (K)I

1

1

1

2

= IJ (K)I [K, J (K)]- 21J (K)1 3K IJ (K)I J (K) = 0

since [K, J (K)] = 0 and K IJ (K)1 2 = O. Hence (r,O) defines local coordinates on a neighborhood of x. Define f (r) ~ IKI (!3r) , where we are using the fact that IKI is constant on !3r. The metric is given by / J (K)

9 = \ IJ (K)I' =

J (K))

2

/

IJ (K)I dr + 2 \

J (K) ) K, IJ (K)I drdO

+ (K, K) dO

2

dr 2 + f (r)2 d0 2.

o It can be shown on a surface that the integral curves of a Killing vector field J (\7 f) are closed loops. In particular one can prove the following without using the Uniformization Theorem (see [86]).

THEOREM 1.19 (Shrinking surface soliton is spherical). If (M2,g) is a shrinking gradient soliton on a closed surface, then 9 has constant positive curvature. More generally, if (M2, g) is a shrinking gradient soliton on a closed, orient able 2-dimensional orbifold with isolated singularities, then 9 is rotationally symmetric with positive curvature. When the 2-orbifold is bad, i.e., not covered by a smooth surface, a unique Ricci soliton exists, which is a nontrivial shrinking gradient Ricci soliton (see [180] and [372]). Topologically, a closed, bad 2-orbifold has either one or two singular points [343]. REMARK 1.20. There do not exist complete shrinking Ricci solitons on noncompact surfaces.

3.2. Warped products. Many known examples of gradient Ricci solitons have been constructed in the form of warped products. In particular, we consider a metric on the product Mn+1 = I x Nn of the form (1.37)

3.

WARPED PRODUCTS AND 2-DIMENSIONAL SOLITONS

13

where r is the standard coordinate on an interval I c JR, 9 is a given metric on an n-dimensional manifold N, and w(r) > 0 is the warping function, which scales distances along the N-factors in the product. Several well-known metrics may be expressed as warped products, at least on an open subset. When N is the unit circle S1, with 9 = d(P for the standard coordinate e, then w(r) = r, w(r) = sinr, w(r) = sinhr give, respectively, the Euclidean plane, the round sphere (with Gauss curvature +1), and the hyperbolic plane (with Gauss curvature -1) with the origin omitted. When N = sn with the standard metric 9 = gcan of constant curvature +1, then w(r) = r gives the standard (flat) metric on JRn+1 with the origin omitted. In each of these cases, the metric extends smoothly across the origin. When N = sn, Lemma 2.10 on p. 29 in Volume One (Le., Lemma A.2 of this volume) gives sufficient conditions for smoothly closing off the Riemannian manifold M, by a point at one (or both) ends of the interval I. Namely, assuming we wish to close off as r ~ 0, we need limr-+o w(r) = 0 and limr-+o w'(r) = 1 (the prime denotes the derivative with respect to r). For constructing Ricci solitons, it is convenient that 9 be a sufficiently 'nice' metric on N; for the rest of this section, we will assume that 9 is an Einstein metric, with Rc(g) = pg. An easy calculation in moving frames gives the following (see Proposition 9.106 on p. 266 of Besse [27]). LEMMA 1.21 (Ricci tensor and Hessian of warped product). If 9 is an Einstein metric on a manifold Nn, with Einstein constant p, then the Ricci tensor of the warped product metric (1.37) is given b'lf (1.38)

w"

Rc(g) = -n-dr2 W

+ (p -

ww" - (n - 1)(w')2) g.

Furthermore, if f is any function of the radial coordinate r, then the Hessian of f with respect to 9 is given by (1.39)

"V"Vf

=

j"(r)dr2 +ww'f'g.

J:

For example, if f(r) = w(t)dt, then "V "V f = w'(r)g. Conversely, by an argument of Cheeger and Colding, warped products may essentially be characterized by the existence of a function whose Hessian is some function times the metric; see §1 in [71] for details. 3.3. Constructing the cigar and other 2-dimensional solitons. For the sake of starting with something familiar, before we study the Bryant soliton, we will briefly repeat the construction of the cigar metric, which is discussed at greater length in Chapter 2 of Volume One. Suppose we wish to construct a complete, steady, rotationally symmetric gradient soliton metric on JR 2. Such a metric will be a warped product (1.37), 5In the case where (N, g) is the unit n-sphere, this follows from (1.58).

1. RICCI SOLITONS

14

where N = Sl, and it is natural to assume that Rc(g) + 'V 2 f = 0 for a radial function f(r). Using (1.38), (1.39) with p = 0 and n = 1 gives (1.40) Integrating

wi" - w" = 0 = w(w' f' - w").

wi" - w' f' = 0 gives

f'

(1.41)

=

for some constant a, whereupon w' f' (1.42)

2aw

- w" integrates to

w' - aw 2 = b.

Using the closure conditions w(O) = 0, w'(O) = 1, we get b = 1. Integrating, we obtain a smooth odd function w(r) whose type depends on the sign of a: • for a = 0, w(r) = r, giving the fiat metric; 1 • for a = a 2 , w(r) = -tan(ar); a 1 2 • for a = -a , w(r) = -tanh(ar). a The third case is the cigar soliton (see [180], [373]) (1.43)

gcig =

dr

212

+ 2" tanh a

2

(ar)dO ,

where by (1.41) we see that the potential function may be taken to be f(r) = -2Iogcosh(ar). The Gauss curvature is K = 2a 2 sech2 (ar) > O. In the second case, the metric (see [125]) (1.44)

1 2 2 = dr 2 + 2" tan (ar)dfJ ,

gxpd

a o < r < 7r/ (2a) , which we call the exploding soliton, is not complete, since tan( ar) ---. 00 at a finite distance away from the origin. The potential function is f (r ) = - 2 log cos (ar) and the Gauss curvature is K = -2a 2 sec 2 (ar) < O. The above steady gradient solitons are defined on topological disks. By taking b = -1, we have the following steady solitons on the punctured disk. For a = -a 2 , taking w(r) =

!a coth(ar) yields

9 = dr 2

1

2 (ar)d0 2 , + 2"coth a

which has potential f(r) = -2logsinh(ar) and K = -2a 2 csch2 (ar) < O. For a = a 2 , taking w(r) =

!a cot(ar) yields

9 = dr 2

1

2 (ar)d0 2 , + 2"cot a

which has potential f (r) = 2 log sin( ar) and curvature K = 2a 2 cot( ar) > O. Both of the above metrics are incomplete because of their behavior near r = O.

3.

WARPED PRODUCTS AND 2-DIMENSIONAL SOLITONS

Likewise, taking b

15

= 0, we have the ODE w' = aw 2. If a = -0: 2, then

w(r) = 0:2;+C' f(r) = -2 log (0:2r+C) and K=

-(r+C:-2)2'

EXERCISE 1.22. Show that if w is a solution to (1.42), i.e., w' - aw 2 = b, then v ~ 1/w satisfies v' +bv 2 = -a. Related to the above exercise, in Section 9 of this chapter we shall see Buscher duality exhibited in the above solitons. EXERCISE 1.23. Determine all of the solutions of the rotationally symmetric steady gradient Ricci soliton equation on a surface. 3.4. A metric transformation on surfaces related to circle actions. It is interesting to search for duality transformations, besides Buscher duality, for Ricci solitons. One transformation for rotationally symmetric metrics on surfaces is the following, which is related to the study of collapsing sequences of solutions of the Ricci flow on 3-manifolds. Given a rotationally symmetric metric 9 = dr 2 + w (r)2 d(P on a surface M2, we may define another metric on M2 by

9 ~ dr 2 +

w (r 0:2w (r)

r

+1

d0 2.

A more general transformation is given by Cheeger [69J. The inverse transformation is g=dr 2 + w(r)2 d0 2 . 1-0:2w(r)2' which is defined as long as 0 < w (r) < 1/0:. EXERCISE 1.24. Consider the SI action on (M2,g) x SI (1/0:), where SI (p) ~ lR/271pZ, defined by ¢(r,O,"1)~(r,O+¢,"1+¢),

Show that

¢ESI.

9 is the quotient metric on [(M2,g) x SI (1/0:)] /SI.

SOLUTION. See Proposition 2 of [103J. We now consider some examples. The inverse transformation of the cigar, ~ . 2( o:r )dO 2, 9 cig = dr 2 + 21smh 0: is the hyperbolic metric of constant curvature K == -0: 2. The transformation of the exploding soliton, gxpd = A

dr 2 + 21sm . 2( o:r )d02 ,

0: is the spherical metric of constant curvature K == 0: 2. Since gxpd is only defined for 0 < r < 7r / (20:) , we see that 9xpd is a hemisphere. In summary, we have the following types of examples:

1. RlCCI SOLITONS

16

(1) a constant negative curvature metric on a surface transforming to a steady soliton, (2) an incomplete steady soliton on a surface transforming to an incomplete metric with constant positive curvature. 3.5. Classifying 2-dimensional solitons. PROPOSITION 1.25 (Surface solitons conformal to ]R2). The only complete steady gradient solitons conformal to the standard metric on ]R2 are the cigar and the fiat metric.

1.26. We have not assumed the curvature is bounded or has a sign. Note that since a steady gradient soliton is an ancient solution, by applying the maximum principle to the evolution equation for the scalar curvature, one sees that a complete steady Ricci soliton on a surface with curvature bounded from below is either flat or has positive curvature (see [111]). For a similar result to Proposition 1.25, see Corollary 1.28 below. REMARK

Consider the pullback of the steady gradient soliton metric under the map from the cylinder 51 x ]R to ]R2 defined by PROOF.

x = eU cosv,

y = e U sinv,

where (u, v) are coordinates on 51 x ]R and 51 = ]R/27l"Z. The pullback is a steady gradient soliton on the cylinder, and the vector field X that it is flowing along is conformal. Hence, the complexification6 of X is a holomorphic vector field and is of the form h(w){)I{)w, where w = u + iv and h(w) is a 27l"-periodic entire (analytic) function. Since X is the gradient of a function f, then X can have no closed orbits, and hence any zeros of h must be simple (by virtue of appealing to a local power series expansion). Consequently, critical points of f can only be local maxima and minima, so that indexp (\7 f) = 1 at any critical point p. Since X (51 x]R) = 0, the Poincare-Hopf Theorem, which says that the sum of the indices of \7 f is the Euler characteristic, implies that f has no critical points, hence h has no zeros. The periodicity then implies that h is a constant. Moreover, X must be a constant times 01 ou, since otherwise the corresponding vector field X on ]R2 has orbits which spiral into the origin. Hence we know that X = I up to multiple, and then by the argument of subsection 3.1 of this chapter, J(X) = 0100 is a Killing field, and so by Lemma 1.18 the metric on ]R2 is rotationally symmetric (and a warped product). Then we may appeal to the above calculation of solutions 0 to (1.40).

ro or

6The complexification of a vector field X is defined + iJ (X)). For example, the complexification of r =

~ (X

~

gr

to

xx 88

+

be y

gy

X(l,O)

is

Z

gz

(x :x + y gy + i (Y go; - xgy )) . See the next chapter for a more detailed discussion of

Kahler manifolds (real surfaces are I-complex dimensional Kahler manifolds). Since we are on a surface, the complexification of a conformal vector field is a holomorphic vector field.

4.

CONSTRUCTING THE BRYANT STEADY SOLITON

17

We have the following from Theorem 13 on p. 61 and Theorem 10 on p. 55 of Huber [210] (see also §10 of Li [251]). THEOREM 1.27 (Conformal structure of surface with finite total curvature). If (M2,g) is a complete Riemannian surface with fM K_dl1 < 00, where K_ ~ max {-K,O} , then (M,g) is conformal to a closed Riemannian surface with a finite number of points removed. Furthermore, by the Cohn- Vossen inequality

1M K+dl1 ~ 1M K- dl1 + 27rX (M) <

00,

where K+ ~ max {K,O} . By the first part of the above theorem, a complete noncom pact surface with positive curvature, which we know is diffeomorphic to the plane, must be conformal to the plane. From this and Proposition 1.25 we conclude the following.

°

COROLLARY 1.28 (Steady surface soliton with R > is cigar). If (M2, g) is a complete steady gradient Ricci soliton with positive curvature, then (M, g) is the cigar. This result gives us a classification of complete steady Ricci solitons on surfaces with curvature bounded from below since such solutions are either flat or have positive curvature.

4. Constructing the Bryant steady soliton We may generalize the cigar metric to a rotationally symmetric steady gradient Ricci soliton in higher dimensions on ]Rn+l by setting N = sn, the unit sphere with constant sectional curvature +1. 7 As the following calculations parallel unpublished work of Robert Bryant for n = 2, we will refer to the complete metrics obtained as Bryant solitons. The Bryant soliton is a singularity model for the degenerate neckpinch, a finite time singularity which is expected to form for some (nongeneric) initial data on closed manifolds. (See Section 6 in Chapter 2 of Volume One.) REMARK 1.29. A singularity model is a long existing solution (Le., the time interval of existence has infinite duration) of the Ricci flow obtained from a singular solution (Le., a solution defined on a maximal time interval [0, T)) of the Ricci flow as the limit of dilations about a sequence of points and times approaching the singularity time T. 7For background on Einstein metrics which are warped products over I-dimensional bases, see the notes and commentary at the end of this chapter.

1. RICCI SOLITONS

18

4.1. Setting up the ODE for Bryant solitons. With p = n - 1, substituting (1.38) and (1.39) into the steady gradient soliton condition Rc(g) + \1\1 f = 0 gives the following system of ODE for wand f:

" !" = n~, w

(1.45) Since R !:If -

=

WW ' f'

= ww" -

(n - 1)(1 - (w ' )2).

-!:If for a steady gradient soliton, Proposition 1.15 implies that

1\1 fl2

= C, a constant. On the other hand, tracing (1.39) gives

!:If

f'

=!" + n~. w I

Using this expression for !:If, we obtain w 2!"

Eliminating (1.46)

+ nww' !, -

w 2 (I') 2

= Cw 2.

f" and w" using (1.45) gives a first integral of our ODE system: 2nww' !, + n(n - 1)(1 - (W')2) - w 2(f')2 = Cw 2.

The analysis of solutions to (1.45) is simplified by using variables that are invariant under the symmetries of the system (Le., translating r, translating f, and simultaneously scaling rand w). We choose new dependent variables x and y and independent variable t (not to be confused with time), such that •

X'T

• I -w f' , Y'Tnw

I

W ,

dr

dt~-.

w

From (1.46) we have nx 2 - y2

(1.47)

+ n (n -1) = Cw 2.

Then by (1.45)

dx

2

dt = ww" = ww' !, + (n - 1)(1 - (w') ),

dy dt Thus the

ODE

I

= wy =

I

I

-ww f .

system (1.45) becomes

dx

dy dt = x(y - nx). ' Substituting C = 0 in the first integral (1.47) gives an invariant hyperbola y2 - nx 2 = n(n - 1) for the system (1.48); see Figure 1 on the next page. (1.48)

REMARK

-

dt

1.30. When n = 1, equation (1.48) becomes dx dt = x (x - y),

(1.49)

which implies x

~; = 2x (x -

2

=x -xy+n-1

+y =

dy dt = x(y - x),

const. Assuming, x, y

~

1 as t

~ -00,

we have

1) . The solution of (1.49) with x (t) decreasing is given by 1

x

= 1 + e2t '

1

y

= 2 - 1 + e2t .

4.

CONSTRUCTING THE BRYANT STEADY SOLITON

19

FIGURE 1. Phase portrait for system (1.48), n = 2, drawn using Maple.

Since ftlogw

= x,

we have w

r

=

= (e- 2t + 1)-1/2.

J

wdt

= arcsinh

= In ( et + J 1 + e2t ) (e t ) .

Hence et = sinh r, so that w (r) = (csch 2r the cigar soliton. EXERCISE 1.31. For n and x (t) increasing?

=

Moreover

r 1/2

+1

= tanh r, and we have

1, what is the solution with x, y

~

1 as t

~

-00,

We will establish the existence of a complete steady gradient Ricci soliton on ]Rn+l by doing phase plane analysis on the system (1.48). Note that the stationary solutions of (1.48) satisfy y = nx and x 2 - 1 = 0, so that the stationary points are (x, y) = (1, n) and (x, y) = (-1, -n). These are both

20

1. RICCI SOLITONS

saddle points since the linearization of (1.48) at (1, n) is8 du dt = - (n - 2) u - v, dv

- = -nu+v dt

'

and the negative of this at (-1, -n) Since we want the metric to close up smoothly as r ---t 0, by Lemma A.2, we will need

.9

x

---t

1 and

y

---t

n.

Therefore, we will limit our attention to the two trajectories that emerge from the saddle point (1, n) as t increases. (These, plus the point itself, comprise the unstable manifold of the saddle point; the stable manifold is the hyperbola.) Given either of these trajectories, we can reconstruct functions w, rand f of t by successively integrating dw (1.50) - = x dt, dr = w dt, df = (nx - y)dt. w Furthermore, we can choose constants of integration such that w '\. 0, r '\. 0, and df /dr ---t 0 as t ---t -00. It then follows that f(r) and w(r) smoothly extend to even and odd functions of r, respectively (cf. [217], Proposition 5.2). 4.2. Phase plane analysis of the right-hand trajectory. The linearization of (1.48) at the saddle point shows that the right-hand trajectory emerges from the saddle point into the region where dx / dt > 0 and dy / dt < o. Because trajectories along the boundary of this region only point into the region, x continues to increase and y continues to decrease. Linearization also shows that y - nx is initially negative along the trajectory, and the equation d dx -(y - nx) = x(y - nx) - n dt dt shows that y - nx is negative and monotone decreasing. Because x ~ 1 and y - nx < -C for some positive constant C when t is sufficiently large, 8The linearization of the general system dx dt = f(x,y),

dy dt=g(x,y)

at (x,y) is di dt

= ax (x,y)x+

of

_

of _ oy (x,y)y,

djj dt

= ax (x, y) x + oy (x, y) y.

og

_

og

_

9The eigenvalues of the matrix

are 2 and 1- n, with corresponding eigenvectors (-l,n) and (1,1), respectively.

4. CONSTRUCTING THE BRYANT STEADY SOLITON

21

the differential equation for Y in (1.48) shows that there exists to such that Y :::; 0 for t 2: to. Therefore,

dx 2 ->x +n-1 dt and limt/T x =

+00 for some finite T > to.

PROPOSITION

1.32. The metric associated to this trajectory is incom-

plete. PROOF.

(1.51)

We will show that there exists a constant a E (0,1) such that a x<-- T-t

for t sufficiently close to T. From this inequality, it follows by integration that w :::; C (T - t) -a for some constant C and hence that r = w dt is finite as t /' T. To establish the upper bound on x, suppose we knew that there is a positive constant 0 and a tl E (t, T) such that

J

(1.52)

y

Then dx/dt 2: (1

+ Ox :::; 0 for

+ 0)x 2 + n -

d -arctan dt

all t 2: tl.

1, and this implies that

(x) > ~-1 -Ja(n-1) a'

a ~ 1/(1 + 0).

Integrating from t to T on both sides and solving for x give the inequality (1.51). To establish (1.52), note that d dt (y

+ ox) = x(y - nx) + 0(x 2 - xy + n -

1).

When y + Ox = 0, the right-hand side equals (0 2 - n)x 2 + (n - 1)0. Let Xo be the x-intercept of the trajectory, and let 00 > 0 be small enough so that (1.53)

(05 - n)x 2

+ (n -

1)00 < 0 for all x > Xo.

Let (Xl, Yl) be any point on the trajectory such that and choose 0 :::; 00 sufficiently small so that Yl + OXI :::; that this inequality persists for all later values of t.

Xl

o.

> Xo and Yl < 0, Then (1.53) shows D

4.3. Phase plane analysis of the left-hand trajectory. We now turn to the analysis of the other trajectory emerging from saddle point (1, n). By the same reasoning as before, we see that x is decreasing, Y is increasing, and Y - nx is positive along this trajectory. Moreover,

x

---4

0+

and

Y ---4

+00

as t increases. In order to prove that the corresponding metric is complete, we need to know the limiting behavior of wand r. To do this, we introduce

22

1. RICCI SOLITONS

new phase plane coordinates

Y~ y'n(n-l).

x X ~ y'n-, y As x

---t

0+ and y

---t

y

+00, X

---t

0+

and

Y

---t

0+.

(The constants are chosen so that the invariant hyperbola becomes the unit circle in the XY-plane.) With a new independent variable s such that ds ~ y dt, we have

~~ = X 3 -

(1.54)

X

~: = Y

+ ay2,

(X2 - aX) ,

where a ~ 1/ y'n. In these coordinates, the first integral corresponds to a Lyapunov function L ~ X2 + y2 which satisfies dL/ds = 2X2(L -1). Since L < 1 for this trajectory, L is strictly decreasing but dL/ds 2 2L(L - 1). So, the trajectory approaches the origin exponentially as s ---t +00. Since (1.55)

= xdt =

d(logw)

1 dY y'nX -1 Y'

it suffices to find the limiting behavior of X in terms of Y. Note that for X and Y small, (1.54) implies dX 2 r::::: -X+aY ds '

dY

ds

r:::::

as s

---t

-

In particular, we expect -X following. LEMMA

+ ay2

---t

°

-aXY.

+00. Indeed, we have the

1.33.

and

where a =

l/vin.

PROOF. One might try to apply I'Hopital's Rule to find the limit of X /y2. However,

(1.56)

2

tsx _~ (X2X2- - aX1 ) +

.!l.. y2 - y2 ds

1 'nX2 - X

v'·

shows that this is not straightforward, since the limit of X/Y 2 is also involved in the right-hand side. Let g(s) = X and h(s) = y2 and apply the Cauchy mean value formula (see, e.g., [10]):

g'(c)(h(a) - h(b)) = h'(c)(g(a) - g(b))

4. CONSTRUCTING THE BRYANT STEADY SOLITON

23

for some c between a and b. Letting b --t +00 gives g'(c)jh'(c) = g(a)jh(a). Substituting in from (1.56) gives

~ (x~2_-a~) + v'nX ; - X Is=c =

2:

2

Is=a'

Multiplying through by X 2 - aX (evaluated at c) and taking a (which forces c --t +00) give

X . 11m y2 +a

s ...... +oo

--t

+00

= O.

However, to get zero on the right-hand side, we need to first know that Xjy 2 is bounded as s --t +00. We know that Xjy2 2 O. To obtain an upper bound, we use the differential equation :s

(~)

=a -

~ (X2 -

2aX + 1)

.

Since the trajectory approaches the origin in the XY-plane exponentially, there exists a k > 0 such that d (X) ( -X - a ) (e-k s -~ -~ < - ) 1 + ae- ks ~ .

Comparing with the solution of the

ODE

dujds = u(e- ks - 1) + ae- ks

shows that Xjy 2 is bounded above. In particular, for s 2 So

~ (s) ~ u (s) = e-(ke-kS+ s)

(1:

eke-kS+Sae-ksds + e-k

~ (so))

.

The second limit, which gives the next term in the power series expansion of X in terms of Y near the origin, follows by applying the same arguments to the equations

d (X -y4a y2 )

ds

(

2

= -1 - 3X + 4aX

)

(X _y4aY2)

- ~ y2 (X2 -2aX)

and

o EXERCISE

1.34. Verify the second limit.

24

1. RICCI SOLITONS

Now we derive the asymptotic behavior of wand r. We have d X dy(logw+logY) = Y(X-a)'

which by Lemma 1.33 approaches zero as Y ~ O. Hence, wY approaches some positive constant G as Y ~ O. Next, note that dr

dY

w

vn(n - 1)(X2 - aX)

shows that r is monotone increasing as Y decreases to zero, because X - a = a(nx - y)/y is negative along this trajectory. Furthermore, the limiting behavior of X and w implies that there is a constant c such that dr

w

dY

vn(n - l)X (a - X)

-- =

>

G

----===-----

v'n"=1(1 + c)y3

for Y sufficiently close to zero. We can similarly find a constant c' such that dr dY

2(1

+ c')G

--<----';:==~

v'n"=1y3

for Y sufficiently close to zero. Thus, we see by integration that r is O(y-2) as Y approaches zero. In particular, r ~ +00. Therefore, the metric is complete. This proves the following. THEOREM 1.35 (Existence of Bryant soliton). There exists a complete, rotationally symmetric steady gradient soliton on ~n+1 which is unique up to homothety. REMARK 1.36. Since w = 0 (y-l) and r = O(y- 2 ), we have w (r) = o (rl/2) . This tells us the Bryant soliton is like a paraboloid. Throughout this section, by x = 0 (yP) we mean there exists a positive constant G such that G-1yp :::; x :::; Gyp. This is an abuse of notation since usually x = 0 (yP) means Ixl :::; GyP for some G. 4.4. Geometric properties of Bryant solitons. It is interesting to contrast the asymptotic behavior of the curvature for these metrics with that of the cigar metric. Recall from Chapter 2 of Volume One that the Gauss curvature of the cigar falls off exponentially as a function of the distance to the origin; in fact, since w(r) is asymptotically constant as r ~ 00, the cigar is asymptotic to a cylinder. On the other hand, in a higher-dimensional warped product, the n-dimensional volume of the sphere at distance r from the origin is w(r)nvol(sn), where vol(sn) is the volume of the standard nsphere. Since w(r) = O(rl/2), we see that the sphere volume is unbounded. From the limits in Lemma 1.33, we obtain the asymptotics of the derivatives of w as functions of r: Wi

= X = v'n -

1 X/Y

= O(Y) = O(r-l/2)

25

4. CONSTRUCTING THE BRYANT STEADY SOLITON

and

w"

l'

~

-

xy +n -1 w

= 0(_y3) = 0(_1"-3/2)

(1.57)

as

x2

00.

A moving frames calculation like that mentioned above lO shows that the sectional curvatures of the Bryant solitons are (1.58)

VI

=

1- (w'? W

V2

2'

w"

= --, w

where VI is the curvature for planes tangent to the spheres, and V2 for planes tangent to the radial direction. In particular, 2VI is the eigenvalue of the curvature operator corresponding to the eigenspace El ~

{Q 1\ ,B : Q,,B E AI (sn

X

{1"})}

and 2V2 is the eigenvalue corresponding to the eigenspace

E2 ~

{Q 1\ dr : QE Al (sn

X

{r})} .

From the asymptotic behavior of wand its first two derivatives, we obtain VI = 0(1'-1) and V2 = 0(1'-2). So, the sectional curvatures decay inverse linearly in terms of the distance from the origin. Like the cigar, the curvature is positive: LEMMA 1.37 (Curvature of Bryant soliton). The curvature operator of the Bryant soliton is strictly positive (i. e., VI > 0 and V2 > 0) away from the origin, and these curvatures have a positive limit at the origin. Moreover, VI

= 0(1'-1) and

V2

= 0(1'-2).

PROOF. The positivity of VI for l' > 0 follows from the fact that w' = x is strictly less than 1. To show that V2 is positive for l' > 0, we will show that w" is negative. From (1.57), it suffices to show that x 2 - xy + n - 1 is negative, or equivalently that X 2 - y'riX + y2 is negative. Linearizing the system (1.48) about the saddle point shows that x 2 - xy + n - 1 is initially negative when the trajectory emerges. Moreover,

.!!..-(X2 _ ds

Vn + y2) = X2(X2 + y2 -

1) + (X2 _ 1)(X2 - ynX

+ y2)

shows that if X 2 - y'ri + y2 were ever equal to zero in the middle of the trajectory, it would have a negative derivative. So, w" is negative for all l' > O. At the origin, all sectional curvatures are equal. The scalar curvature R satisfies R + IV' fl2 = C, and we know C must be positive because

R = n(n -

l)VI

+ 2nv2

lOSee, for example, the solution to Exercise 1.188 in [111].

1. RICCI SOLITONS

26

is positive everywhere else. The curvatures approach zero as r ---t +00, so R has a positive maximum value at some point. Because f' = (y - nx)/w is positive for r > 0, this must occur at the origin. D We end this section by noting that the preceding construction can be generalized to the case where the fiber is a product of an Einstein manifold and a sphere (see [219]). THEOREM 1.38 (Steady solitons on doubly-warped products). Given an Einstein manifold (Nm, 9N) with positive Ricci curvature, there exists a 1parameter family of complete steady gradient soliton metrics on ~n+l x N in the form of doubly-warped productsll,' 9 = dr 2 + v(r)2gcan where

gcan

is the standard metric on

+ w(r)2gN,

sn, n ~ 1.

These solitons have positive Ricci curvature when r > O. However, the sectional curvature along the copies of N can be negative.

5. Rotationally symmetric expanding solitons In Section 4 in Chapter 2 of Volume One, we described the construction of a I-parameter family of rotationally symmetric expanding gradient Ricci solitons on ~2 (see also Gutperle, Headrick, Minwalla, and Schomerus [174]). These solitons are asymptotic to a cone (with any cone angle in the interval (0,27T') possible) and have curvature exponentially decaying as a function of the distance to the origin (see Exercise 4.15 and Corollary 9.60 of [111]). In this section we consider the problem of constructing rotationally symmetric expanding gradient Ricci solitons on ~n+l for n ~ 2. In particular, we generalize the construction from the previous section on the Bryant soliton to show that there exists a I-parameter family of rotationally symmetric complete expanding gradient solitons on ~n+l.

5.1. Setting up the ODE for expanding gradient solitons. To start, we begin with the expanding gradient soliton condition

Rc(g)

+ VV f + Ag =

0,

E

A = "2 > O.

Substituting (1.38) and (1.39) into this equation gives

o=

-n~" dr 2 + (p - ww" - (n - I)(w')2) 9 w

+ (I" dr 2 + ww' fig) + A (dr2 + w 2g) , is the warping function and 9 is the metric on sn with Einstein

where w( r) constant p. Taking p = n - 1 and collecting the components of the metric

llThis construction was inspired by a similar construction for Einstein metrics by Berard-Bergery [24].

5. ROTATIONALLY SYMMETRIC EXPANDING SOLITONS

on jRn+1, we get a system of two second-order of r:

f" + >. = nw" /w,

ww' f'

ODES

for

27

f and w as functions

+ >.w 2 = ww" + (n -

1)((w' )2 - 1).

Again, Proposition 1.15, together with equation (1.25), provides a first integral for this system:

f" + n(w' f' /w) -

(f')2 - 2>.f = C.

We can reduce the order of the system by again introducing variables that are invariant under the symmetries of translating r and translating f:

Y =;=• nw I - w f' .

. I x=;=w,

Because simultaneous scaling of rand w is no longer a symmetry of the system, we must also retain w as a variable, yielding the following firstorder system:

= xw, dx/dt = x 2 - xy + >.w 2 + n -1, dy/dt = x(y - nx) + >.w 2 ,

dw/dt (1.59)

where the parameter t is related to r by dt = w- 1 dr. (Notice that the first integral is not invariant under the translation symmetries, so it does not give an invariant manifold for this system.) Clearly, solutions of the steady soliton system (1.48) are also solutions of (1.59) for which w = O. (Of course, in the steady case, the warping function is not identically zero, but it is recovered by the quadrature d(log w) = x dt.) To get a metric that closes up smoothly at the origin, we need a trajectory that emerges from the singular point P = (0,1, n) as r increases. (We take w,x,y, in that order, as coordinates on the phase space JR.3.) Linearization at P shows that there is a 2-dimensional unstable manifold passing through P, whose tangent space at P is spanned by the vectors (1,0,0) and (0, -1, n)P The trajectory corresponding to the steady soliton lies in this surface and is tangent to (0, -1, n) at P. Of course, we are only 12The linearization of (1.59) at P is diD

_

ill =W, dx

dt

= (2 -

d-y

= -nx- +-y.

dt

n) x - ;ii,

Note that the last two equations appeared in the linearization for the steady soliton equation and the eigenvalues of the matrix

-1) ( 2-n -n 1

are 2 and 1- n, with corresponding eigenvectors (-l,n) and (1,1), respectively.

28

1. RICCI SOLITONS

interested in solutions in the half-space where w is positive. There is a 1parameter family of these trajectories in the unstable manifold, and they are tangent to the vector (I, 0, 0) as they exit from P. (This vector is associated to the positive eigenvalue closest to zero.) Among these is the hyperbolic metric, for which f' = 0, y = nx, and the warping function is w

= Vnj>.sinh (V>.jnr) ,

y

= ncosh (V>.jnr) .

5.2. Analysis of a I-parameter family of trajectories. For the rest of this section, we will consider only the I-parameter family of trajectories that lie in the quadrant of the unstable manifold of P, between the hyperbolic metric and the steady soliton, i.e., those for which wand y - nx are positive for small values of r. Among these is the flat Gaussian soliton on lR.n +1 described in Example 1.9; the corresponding solution of (1.59) is

w = r,

x = 1,

y=n

+ >.r2.

In order to show that the metrics corresponding to our family of trajectories are complete and to study their curvatures, we again introduce rescaled coordinates

w W::§=-,

. vn(n - 1)

x X::§=vn-,

y

y=;=

y

y

We introduce a new independent variable s such that ds system (1.59) becomes

.

= y dt, whereupon

d: = W(X2 _ >'W2), dX ds = X 3

(1.60)

-

X

~~ = Y(X2 -

+ ay2 + >.( vn -

X)W2,

a::§= Ijvn,

aX - >'W2).

In these coordinates, P = (0, a, VI - ( 2 ), and the hyperbolic trajectory has X identically equal to a, approaching the critical point H = (Ij-Jri"X, a, 0) as r --t 00. The trajectories we are considering lie in the unstable manifold of P, and for small r, in the region where X < a and W > o. The flat trajectory satisfies Y = v'11=1 X and X 2 - aX + >.W 2 = O. LEMMA

1.39. These trajectories remain in the region defined by

0< W:S Ij~.

0< X < a, PROOF.

Along the plane where X = a,

dXjds = a 3

-

a

+ ay2 + >'(vn -

a)W2.

Let Q stand for the quantity on the right, which must be negative for r small. The following equation shows that Q remains negative if X < a:

dQjds = aXy2(X - a)

+ >.( vn -

a)(X2 - ( 2)W2 - >.QW 2.

5.

ROTATIONALLY SYMMETRIC EXPANDING SOLITONS

29

This establishes that X < a always along these trajectories, and the equation dX/ds = ay2 + AVnW 2 when X = 0 shows that X remains positive. Let L = X 2 + y2. Then dL/ds

shows that L

~

= 2(L -

1)(X2 - AW2)

+ 2AVnW2(X -

1 is preserved. Finally, when X

a)

< a,

dW/ds ~ W(a 2 - AW 2),

showing that

W cannot exceed a/../'X = 1/vn:>...

D

LEMMA 1.40. As s ~ +00, these trajectories approach the origin in W XY coordinates, and the corresponding metrics are complete. PROOF. Because the right-hand sides of the system (1.60) are polynomial, parameter s is unbounded along these trajectories. Because the trajectories are bounded within the region given by Lemma 1.39, each trajectory must limit to either the critical point H or to the origin 0 as s ~ +00. However, linearization shows that the tangent space of the stable manifold of H intersects the closure of the region given in Lemma 1.39 only in the line through H tangent to the hyperbolic trajectory. Thus, all the trajectories under consideration limit to the origin. To show that the metrics corresponding to these trajectories are complete, note that (1.61 )

J J dr =

W ds =

J

X2

~~W2 .

Along these trajectories, W must eventually become a decreasing function of s. Then it suffices to show that there is a point (Wo, Xo, Yo) along one of our trajectories such that the limit

fWO

!~Je

dW AW2 - X2

is infinity; this follows from AW2 - X2 < AW2.

D

The flat trajectory divides our chosen quadrant of the unstable manifold near S into regions of negative sectional curvature (bordering the hyperbolic trajectory) and regions of positive sectional curvature (bordering the steady soliton trajectory). The following lemma shows that these signs persist; thus, there exist I-parameter families of rotationally symmetric expanding solitons of strictly positive and of strictly negative sectional curvature. PROPOSITION 1.41. Excluding the fiat metric, the sectional curvatures of these metrics are either strictly positive or strictly negative for all r.

30

1. RICCI SOLITONS

PROOF. VI =

In these coordinates, the sectional curvatures are

y2 - (n - 1)X2 1 - (W')2 = -----;---'------:-~;o__ w2 n(n - 1)W2 '

for 2-planes tangent to and perpendicular to, respectively, the orbits of the rotational symmetry. The evolution equations for the numerators are 1d 2 --(Y - (n - l)X 2 ) 2 ds

= (X 2 -,XW 2 -

aX)(Y 2 - (n - l)X 2 )

+ a(n -l)X( _(X2

+ y2 + n,XW2 -

vnX))

and 1d 2 r:::: --(-(X +Y 2 +n,XW 2 -ynX)) 2ds = _(X2 - ,XW2 + ~(aX - 1))(X2

+ y2 + n,XW 2 - vnX) a (Y2 - (n - l)X 2) . + "2X

Observing that the sign of the first nonconstant factor in the second line of each equation is positive, we conclude that the signs VI > 0, V2 > 0 and VI < 0, V2 < 0 are both preserved. 0 We will now obtain the asymptotic behavior of the curvature relative to distance r. PROPOSITION 1.42. Assuming that the sectional curvatures are negative, then they both decay at least quadratically in r, and for large r, the warping function w(r) is bounded between two linear functions of r. PROOF. First, we establish that, as the trajectories approach the origin, the distance r is asymptotic to a constant times l/W. We will do this by establishing a positive lower bound for W 2 / X, by examining the equations

~ ds

(W2) = W 2 (X2 _ ,XW2 + 1- (ay2

X

X

+ ,Xy'nW2))

X'

~

(Y2) = y2 (X2 _ ,XW2 + 1- 2aX _ (ay2 dsX X

+ ,Xy'nW2))

X·

For r sufficiently large, we can bound the nonconstant terms in the parentheses (those that do not involve dividing by X) by a small number c. Then, if we set Y ~ ay2/X and W ~ 'xy'nW 2/X, we have

:s (W

+ Y) ~

(W

+ Y)

(1 - c -

Y-

W) .

Hence, W + Y ~ c for some positive constant c. On the other hand, because sectional curvature VI is negative, y2 / X ~ (n-1)X, and so lims -++ oo Y = o.

5. ROTATIONALLY SYMMETRIC EXPANDING SOLITONS

31

Hence W 2 / X 2: C1 for some positive C1, for r sufficiently large. Thus, the denominator in the integral (1.61) is bounded below, 2

2

AW - X

W4

2

2: AW - (C1)2'

and it follows that r = O(I/W). The orbital sectional curvature satisfies 111

=

y2-(n-l)X2 X2 W2 > --- > --n(n - I)W2 nW2 n(ct)2

and so decays at least as fast as 1/r2. The equation 112

(1.62)

+ (n -

1)111

=-

(X2 - aX + AW2) W2

shows that in order for 112 to also decay to zero, it is necessary that lim X/W 2 s---+oo

(1.63)

=

AVn.

We already know that X/W 2 is bounded above, and we can then apply the Cauchy mean value theorem (as in Lemma 1.33) to the equation 2

.!LX ds

-

dW2 -

~

(X2 - AW 2 + a~

-1) + AVn

-----'-------;::-------:--::-::-,.".---''----

X2 -AW2

ds

to obtain the desired limit. Next, dividing (1.62) by W 2 gives -112 111 W2 = (n - 1) W2

X2

+ W4 +

A - a~ W2 .

In order to show that 112 decays at least as fast as 1/r2, we need only show that the last term is bounded. This follows from the differential equation

:. C-:l) ~(_1+2,\W2 + A(X -

-3X2)

(,,-:l)

X

y2

a) W2 - a 2 W4'

where the boundedness of the terms on the second line follows from the limit (1.63) and the negative curvature condition y2 ~ (n - I)X2. To establish the last assertion, it suffices to show that dw / dr = x = v:n=T X/Y is bounded above and below for r sufficiently large. First, because of (1.62) we can say that -d (W2) = (X2

ds

Y

+ aX -

W2 AW2)-

Because of the limit (1.63), d W2 - l o g - < CW4 ds Y-

W2 < 2X2_.

Y-

Y

32

1.

RICCI SOLITONS

for some constant C > O. Because dr = W ds and W = 0 (1/ r ), then d W2 - l o g - < Cr- 3 dr Y ,

from which it follows that W 2 /Y is bounded above for large r. Thus, there are positive constants Cl, C2 such that

for sufficiently large

T,

and the bounds on X/Y follow.

o

Much more detailed information about the solutions in the positive and negative curvature cases (as well as the steady case) has been obtained by Robert Bryant (in unpublished notes, using different coordinates). In particular, Bryant proved PROPOSITION 1.43 (Bryant). Each positive curvature solution defines a complete, rotationally symmetric expanding soliton, whose sectional curvature decays proportionally to 1/r2. 6. Homogeneous expanding solitons 6.1. Existence. Homogeneous spaces are among the nicest examples of Riemannian manifolds. 13 Einstein metrics are among the nicest examples of Riemannian metrics. It is thus natural to ask if a given homogeneous space M n = G / K admits a G-invariant Einstein metric. If M is closed, the answer is frequently 'yes'. For example, consider the following result of B6hm and Kerr [29].

THEOREM 1.44. Let Mn be a closed, simply-connected homogeneous space. If n < 12, then M admits a homogeneous Einstein metric. On the other hand, Wang and Ziller's example SU(4)/SU(2) of a 12dimensional homogeneous space that admits no homogenous Einstein metric shows that the answer is 'no' in general and that the dimension restriction above is sharp [365]. Every known example of a noncompact, nonfiat homogeneous space that admits an Einstein metric is isomorphic to a solvable Lie group S. (See [198] and [317].) Moreover, for all known examples, the Einstein metric is of standard type. DEFINITION 1.45. A left-invariant metric g on a solvable Lie group S, regarded as an inner product on the Lie algebra.5, is said to be of standard type if the orthogonal complement with respect to g of the derived algebra [.5,.5] forms an abelian subalgebra a of .5. 13See Chapter 7 of [27], for example.

6. HOMOGENEOUS EXPANDING SOLITONS

33

Many noncom pact homogeneous spaces admit no Einstein metrics whatsoever. For instance, there is the following result of Milnor [266, Theorem 2.4]. THEOREM 1.46. If the Lie algebra of a Lie group N is nilpotent but not commutative, then the Ricci curvature of any left-invariant metric on N has mixed sign.

The Ricci soliton structure equation (1.64)

Rc

=

-Ag - £Xg

illustrates the sense in which a Ricci soliton may be regarded as a generalization of an Einstein metric. Therefore, it is natural to look for Ricci solitons on homogeneous spaces, such as nonabelian nilpotent Lie groups, that do not admit any Einstein metric. Existence and uniqueness of Ricci solitons on such spaces has been investigated by Lauret [244]. To describe his results, we must recall some notation from the cohomology of a Lie algebra g, relative to its adjoint representation. (See [232] and [369].) A k-cochain is a skew-symmetric k-linear map 9 x ... x 9 --t g. Denote the vector space of all k-cochains by C k , noting the natural identifications CO = g, C 1 = End(g) = g* ® g, and C 2 = A2 (g*) ® g. The Lie algebra cohomology of 9 relative to its Chevalley complex,

Hk(g) = ker(8 k )jim(8 k -

1 ),

is derived from the coboundary operators 15k : C k

8k(A)(X 1 , .•• ,XHr)

--t

CHI defined by

,,+k (-1)1 [Xj,A(X 1 , ... ,Xj, ... ,XHl)]

=

~

A

l~j~k+1

+ l~i<j~k+1

In particular, one has 80(X)(Y) = -[Y, X] = adx(Y) for all X, Y E g, and

81 (A)(X, Y)

= [X, A(Y)]- [Y, A(X)]- A([X, Y])

for all A E End(g) and X, Y E g. A derivation of 9 is an element ofker(8r). Let g be a left-invariant metric on a simply-connected Lie group G. Regarding the Ricci curvature Rc of g as an endomorphism, Lauret considers the condition

(A E JR),

(1.65) which is easily seen to be equivalent to (1.66)

Rc

= -AI +D

(A E JR, DE ker(8r)).

Notice that equations (1.65) and (1.66) relate the geometry of (G,g) to algebraic data of g. This turns out to be a productive point of view. We now survey some of the results it generates, referring the reader to [244] for the detailed proofs.

34

1.

RICCI SOLITONS

One begins with the observation that conditions (1.64)-(1.66) are equivalent for a nilpotent group. LEMMA 1.47. Let N be a simply-connected nilpotent Lie group with Lie algebra n and a left-invariant metric g. Then there exist A E lR and X E n solving the Ricci soliton structure equation (1.64) if and only if there exist A E lR and a derivation D E ker(8d solving (1.66). DEFINITION 1.48. A standard metric solvable extension of (n, g) is a pair (5, g), where 5 = aEBn is a solvable Lie algebra such that [', ']slnxn = [', ']n and 9 is an inner product of standard type such that [5,5]s = n =a..l and

9Inxn =

g.

Recalling Definition 1.45, note that for 9 to be of standard type is equivalent to a being abelian. The following result relates Ricci soliton structures on nilpotent Lie groups with Einstein metrics of standard type on solvable Lie groups. THEOREM 1.49. Let N be a simply-connected nilpotent Lie group with Lie algebra n and a left-invariant metric g. Then (N, g) admits a Ricci soliton structure if and only if (n, g) admits a standard metric solvable extension (5 = aEBn, g) such that the simply-connected solvable Lie group (8, g) is Einstein. Although its proof is nonconstructive, Theorem 1.49 is a very useful criterion. For example, any generalized Heisenberg group [25] and many other two-step nilpotent Lie groups admit a Ricci soliton structure. On the other hand, if n is characteristically nilpotent, then N admits no such structure. (See [244] for more examples.) If a Ricci soliton structure does exist on N, it is essentially unique. THEOREM 1.50. Let N be a simply-connected nilpotent Lie group with Lie algebra n. If g and g' are both Ricci soliton metrics, then there exist a> 0 and TJ E Aut(n) such that g' = aTJ(g). REMARK 1.51. If one regards a Ricci soliton structure as a 3-tuple (N, g, X) satisfying (1.64), then uniqueness of X must be understood modulo addition of a Killing vector field. (See Example 1.57 below.) Some partial answers are known regarding the existence of Ricci soliton structures on broader classes of Lie groups. For example, if G is semisimpIe, then condition (1.66) implies condition (1.64). However, (1.66) has no solutions other than Einstein metrics: THEOREM 1.52. A left-invariant metric 9 on a simply-connected semisimple Lie group G satisfies (1.66) if and only if D = 0, hence if and only if g is Einstein. It is not known in general whether or not a noncompact, semisimple Lie group admits an Einstein metric.

6. HOMOGENEOUS EXPANDING SOLITONS

35

Lauret also gives a variational characterization of Ricci solitons [244]. Roughly speaking, this says that any simply-connected nilpotent Lie group (N, g) admitting a Ricci soliton structure is a critical point of a functional that measures, in a certain sense, how far (N, g) is from being Einstein. We now make this precise. It will be more convenient to fix an inner product and allow the Lie algebra brackets to vary. (Compare with Examples 1.63 and 1.64 below.) Specifically, fix an n-dimensional inner product space (n, (-,.)) and let A = A2 (n*) ® n denote the space of all skewsymmetric bilinear forms on n. (Compare to C2 ;2 im( 01) defined above.) Let N denote the subspace of all nilpotent elements of A that satisfy the Jacobi identity. Then N may be regarded as the space of all nilpotent Lie brackets on n. To each p, EN, associate the simply-connected Lie group NJ.t and the left-invariant metric gJ.t on NJ.t determined by (-, .). Notice that N is invariant under the natural action of the group GL(n). Thus the GL(n) orbit of p, E N corresponds to all simply-connected homogeneous nilpotent Lie groups N isomorphic to NJ.t, and the O(n) orbit of p, corresponds to all (N, g) isometric to (NJ.t, gJ.t). The fixed metric (.,.) on n induces a metric (-,.) on A defined by

(p"v) =

L (p,(Xi,Xj),Xk) (V(Xi,Xj),Xk)

i,j,k

when {Xl, ... , Xn} is any orthonormal basis of n. Let

Using the Ricci endomorphism, Lauret defines a functional F : Nl

---t

lR by

and proves THEOREM 1.53. Let p, E N l . Then (NJ.t, gJ.t) satisfies (1.64)-(1.66) (i.e. admits a Ricci soliton structure) if and only if p, is a critical point ofF.

To interpret this result, observe that E(p,) = IRc(gJ.t) - *R(gJ.t)II 2 measures the trace-free part of the Ricci endomorphism, i.e., how far gJ.t is from being Einstein. But for p, E N l , one has

Hence a local minimum p, of F in Nl should correspond to a homogeneous space (NJ.t' 9J.t) that is closest to Einstein among all near by (Nv, gv).

36

1. RICCI SOLITONS

6.2. Construction. The first explicit examples of Ricci soliton structures on Lie groups were constructed by Baird and Danielo [15] and independently by Lott [256]. We discuss [15] in this section and [256] in subsection 6.3, below. Baird and Danielo discovered soliton structures on 3-dimensional manifolds by studying semiconformal maps to Riemannian surfaces. Significantly, their constructions give the first known examples of nongradient soliton structures. We will only describe two of their conclusions, referring the reader to [15] for details of the method.

1.54. Let N denote the 3-dimensional Heisenberg group nil3 . Recall that N may be represented as the group of upper-triangular matrices EXAMPLE

under matrix multiplication. It is a general fact 14 that any simply-connected nilpotent Lie group is diffeomorphic to ]Rn. So give ]R3 its standard coordinates (Xl, X2, X3) and define the frame field

a

a

a

a

Fl - 2 F2 = 2 ( - - Xl-), F3 = 2 - . - OXl' OX2 OX3 OX3 It is easy to check that all brackets [Fi' Fj ] vanish except [Fl' F2] = -2F3, hence that (Fl' F 2, F3) is a nil 3 -geometry frame. 15 The connection I-forms may be displayed as VFIFI VFIF2 ( VF2H VF2F2 VF3Fl VF3F2 Using the dual field

define a left-invariant metric on N by 9 = 4 (wI ® wI + w 2 ® w2 + w3 ® w3) . Recalling the standard formula

(R(X, Y)Y, X) =

~ I(ad X)*Y + (ad Y)* XI 2 3

- 41[X, Y]I

2

-

I

((ad X)* X, (ad Y)*Y)

2 ([[X, Y], Y], X)

-

I

2 ([[Y, X], X], Y),

it is straightforward to compute that Rc(g) = -2(w l ® wI) - 2(w 2 ® w2) + 2(w 3 ® w3). l4The exponential map of a connected, simply-connected, nilpotent Lie group is a diffeomorphism. For instance, see [200j. l5See Volume One, Chapter 1, Sections 3-4.

6.

HOMOGENEOUS EXPANDING SOLITONS

37

Define a vector field

X

1 2

1 1 -X2F2 - (-XIX2 2 2

= --xIFI -

+ x3)F3.

It is then easy to see that

= £Xg + 3g,

-2 Rc(g)

hence that (N, g, X) is a Ricci soliton structure. REMARK 1.55. Because the I-form metrically dual to X is not closed, it follows that X =1= grad f for any soliton potential function f. REMARK 1.56. Compact locally homogeneous manifolds with nil3 geometry occur as mapping tori of Y A

:

T2

--t

T2 induced by A =

(~ ~)

E

8L(2, Z) with k =1= 0. The left-invariant metric 9 is compatible with any compact quotient, but the soliton structure is never compatible with compactification. Indeed, the scalar curvature of 9 is R = -1/2, while every compact Ricci soliton of nonpositive scalar curvature is Einstein. EXAMPLE 1.57. Let S denote the simply-connected 3-dimensional solvable Lie group sol3 = lR >
F2 =

2(e-Xl~+eXl~), 8X2

8X3

F3 =

2(e-Xl~_eXl~). 8X2

8X3

Its bracket relations are [FI' F2J = -2F3, [F2' F3J = 0, and [F3, FIJ = 2F2. 80 (FI' F 2, F3) is a sol3-geometry frame. The connection I-forms may be displayed as

Using the dual field

define a left-invariant metric on S by

9 = 4(w l ® wI)

+ 8(w 2 ® w2 ) + 8(w 3 ® w3 ).

1. RICCI SOLITONS

38

The Ricci tensor of this metric is simply

Rc(g) = -8(w l ® wI). Given any J-l E

X = J-l

[-H -

~,

define a vector field

e-x1x3F2

+ e-X1x3F3] + (1- J-l) [FI -

eX1 x2F2 - eX1 x2 F3].

The computation implies

-2 Rc(g) = £Xg + 4g, and hence (S, g, X) is a Ricci soliton structure. REMARK 1.58. As in the previous example, X potential function f.

f

grad f for any soliton

REMARK 1.59. Compact locally homogeneous manifolds with so13 geometry are mapping tori of T A : T2 -+ T2 induced by A E SL(2, Z) with eigenvalues A_ < 1 < A+. As in the previous example, the soliton structure cannot descend to any compact quotient, because the scalar curvature of 9 is R = -2.

6.3. Type III singularity models. An important reason for studying shrinking or steady solitons is that they can provide valuable information about finite time singularities of Ricci flow. For example, the (Type I) neckpinch singularity is modeled in all dimensions n 2: 3 by the shrinking gradient cylinder soliton (~ x

sn-l, 9 =

ds 2 + 2 (n - 1) gean, X

= grad(s2/4)).

(See Section 5 in Chapter 2 of Volume One and [7, 8].) The conjectured (Type II) degenerate neckpinch (Section 6 in Chapter 2 of Volume One) is expected to be modeled on the Bryant soliton, discussed above. We shall now see that homogeneous expanding solitons can model infinite time behavior of Ricci flow.

t

E

DEFINITION 1.60. A Type III solution of Ricci flow (Mn,g(t)) exists for [0, 00) (Le., is immortal) and satisfies

sup

tl Rm I < 00.

Mx[O,oo)

There are many examples of Type III solutions (e.g., manifolds with 3 ni1 or so13 geometry) that collapse with bounded curvature. As t -+ 00, these examples exhibit pointed Gromov-Hausdorff convergence to lower-dimensional manifolds. (See Sections 6 and 7 in Chapter 1 of Volume One.) For such solutions, it is not possible to form a limit solution (M~, goo(t)) in a naive way. However, Lott has shown that such solutions may have limits, properly understood, which turn out to be expanding homogeneous solitons [256]. We will describe only two of his results, omitting

6.

HOMOGENEOUS EXPANDING SOLITONS

39

many details. To discuss convergence of Ricci flow solutions, it is necessary to anticipate some material from Chapter 3. In particular, we refer the reader to Definition 3.6 for the notion of Cheeger-Gromov convergence in the COO-topology of a sequence of pointed solutions to the Ricci flow. Now suppose that (M n , g(t)) is a Type III solution of Ricci flow. Fix an origin x E M. For each s > 0, there is a rescaled pointed solution of Ricci flow (M,gs(t),x) defined for t E [0,00) by

1 gs(t) = -g(st). s Lott proves the following: THEOREM 1.61. Let (Mn,g(t)) be a Type III solution of Ricci flow. If the limit (M~,

goo(t), x oo ) = lim (M, gs(t), x) s-+oo

exists, then (Moo, goo(t), x oo ) is an expanding Ricci soliton.

In dimension n = 3, one does not have to assume existence of a limit. THEOREM 1.62. Let (M 3,g(t)) solve Ricci flow on a simply-connected homogeneous space M3 = GIK. Here, G is a unimodular Lie group and K is a compact isotropy subgroup. Then there exists a limit

which is an expanding homogeneous soliton on a (possibly different) Lie group.

Note that the diffeomorphisms with respect to which the convergence of Definition 3.6 occurs become singular as s ~ 00. We will illustrate the content of Theorem 1.62 by relating it to Examples 1.54 and 1.57 (isometric versions of which were also discovered independently by Lott). EXAMPLE 1.63. Let (N, g, X) denote the nil 3 Ricci soliton structure of Example 1.54. With respect to the frame f3 = (Fl' F2 , F3) constructed there, one may regard the fixed metric 9 as the matrix g{3 =

(04 04 0)0 . 004

Now consider the time-dependent frame a(t) = f3A(t) given by

A(t) = (

~ -2a (t) 2

o a(t)

o

a(t))

o , o

a(t) =

J

1 t12

2/ 3

.

1. RICCI SOLITONS

40

With respect to the frame a(t), one obtains the identification ga(t) =

0 0)

4/ 3 !C (

0

o

1C2/3

0

0

1C2/3

.

The limit soliton metric is

goo(t) = 3tga(t)· EXAMPLE 1.64. Let (8, g, X) denote the so13 Ricci soliton structure of Example 1.57. With respect to the frame f3 = (FI' F2, F3) constructed there, regard the fixed metric 9 as the matrix

gp=

(H ~)

Now consider the time-dependent frame a(t)

A(t)

( 0 _1 0)

= a(t)

o

02 0

0

= f3A(t) a(t)

,

given by

=

a(t)

{ft.

With respect to the frame a(t), one gets the identification

(ICo 0 0) I

ga(t)

=

4

0

I 0

0 I lC 4

.

The limit soliton metric is

goo(t) = 4tga(t)· Lott also discovered a 4-dimensional example [256]. We will describe its (time-independent) Ricci soliton structure, leaving construction of the corresponding Ricci flow solution as an interesting exercise for the reader. EXAMPLE 1.65. Let N denote the simply-connected 4-dimensional nilpotent Lie group nil 4 with bracket relations

and all other [Fi' Fj ] = O. The frame field defined in standard coordinates (Xl, X2, X3, X4) on

a F1=-a' Xl

a F2 =-a' X2

a F3 =-a' X3

,J

~ (-~4 -F4 -~4 -~4 ~F2 F3) . 2

0

- F2

0

Fl - F3

by

a a a F4 =Xl+X2+-a a X2 a X3 X4

realizes these relations. The connection I-forms are

(VpF-) =

]R4

F2

PI

0

7.

WHEN BREATHERS AND SOLITONS ARE EINSTEIN

41

Using the dual field define a left-invariant metric on S by

9

= wI

®

wI

+ w2 ® w2 + w3 ® w3 + w4 ® w4 .

Its Ricci tensor is

1 Rc(g) = __ (wI ® wI)

2 Define a vector field

X

1 3 + _(w ® 2

w3 )

-

(w 4 ® w4 ).

= -2XIFI + (-3X2 + XIX4)F2 + (-4X3 + X2X4)F3

- X4F4.

Then one has

LXg

= _2(wl

® wI) - 3(w 2 ® w2 )

-

4(w 3 ® w3 )

-

(w 4 ® w4 )

and thus

-2 Rc(g) = LXg + 3g. Therefore, (N, g, X) is a Ricci soliton structure. 7. When breathers and solitons are Einstein In this section, we review some of the significant results to date on classifying the breathers and Ricci soliton metrics that exist on a given type of manifold. As noted in the introduction, a shrinking or expanding soliton on a closed manifold will evolve purely by diffeomorphisms under the normalized Ricci flow. More generally, we define a breather for the (normalized or un normalized) Ricci flow to be a solution g(t) for which there exists a period T and a diffeomorphism ¢ such that

g(t + T) = ¢*g(t). So a breather solution is a periodic orbit in the space of metrics modulo diffeomorphisms, as compared to a Ricci soliton, which is a fixed point. The following result is the analogue of Proposition 1.13 for breather solutions to the normalized flow (see [218]). PROPOSITION 1.66 (Steady and expanding breathers are Einstein). Any breather or soliton for the normalized Ricci flow on a closed manifold Mn is either Einstein with constant scalar curvature R ::; 0 or it has positive scalar curvature. PROOF. Under the normalized flow, the scalar curvature satisfies aR 0 2 (1.67) -a = b.R + 21Rcl 2 + -R(R - r), t n o

where Rc denotes the traceless part of the Ricci tensor and r is the average scalar curvature. By compactness in space and periodicity in time, there exists a point p E M where R attains its global minimum R min . Since b.R 2: 0 and aR/at = 0 at p, then (1.67) implies that Rmin(Rmin - r) ::; 0,

42

1. o

with equality only if Rc

= o.

RICCI SOLITONS

Thus, if Rmin < 0, then R is constant at this o

time. By applying the same argument at every point, we get Rc == 0 and the metric is Einstein. Otherwise, assume R min = 0; by (1.67), we know that !1R :-::; 0 at p. Let 0 be the open set where !1R < 0 at this time. If 0 is nonempty, then R can only attain its infimum on 0 (which is zero) at a point on a~. Applying the Hopf maximum principle shows that the outward normal derivative of R must be negative there; but this is impossible because V'R = o there. Thus, 0 is empty and !1R 2:: 0 everywhere. Since R achieves its maximum somewhere, by the maximum principle R is constant (in fact, identically zero), and the metric is Einstein by applying the same argument M~~. 0 REMARK 1.67. Note that a breather with positive scalar curvature on a closed manifold is a shrinking breather. In Chapter 6 we will describe Perelman's result that shrinking breathers on closed manifolds are Ricci solitons. The proof of this uses his entropy formula. In Chapter 5 another proof that steady or expanding breathers are Einstein will be given. It WM shown in Proposition 5.10 of Volume One that the only solitons for the normalized flow on compact surfaces were constant curvature metrics. We will now prove the generalization of this to compact 3-manifolds, by first obtaining a curvature pinching estimate for the sectional curvatures. As a corollary to Theorem A.31, we have the following (see Hamilton [186] and one of the authors [218]). COROLLARY 1.68 (Hamilton-Iveyestimate). Assume the normalization infxEM3 v (x, 0) 2:: -Ion the initial metric, where v (x, t) denotes the smallest eigenvalue of the curvature operator. There exists a continuous positive nondecreasing function 'IjJ : IR -+ IR with 'IjJ (u) lu decreasing for u > 0 and 'IjJ (u) lu -+ 0 as u -+ 00, such that for any solution (M 3 ,g (t)) of the Ricci flow on a closed 3-manifold, we have v 2:: -'IjJ (R). That is, Rm 2:: -'IjJ (R) id,

where id : A2

-+

A2 is the identity.

PROOF. By Theorem A.31, wherever v < 0, we have R 2:: Ivl(log Ivl- 3). In particular, if v :-::; -e6 , then R 2:: ~ Ivllog Ivl. The function f (u) = u log u is increMing for u 2:: 1 Ie and hence has an inverse f- 1 : [-1 Ie, 00) -+ [lie, 00). If v :-::; -e6 , then R 2:: 3e 6 and Ivl :-::; 1-1 (2R) . If we let

'l/J (u)

= { f- 1 (2u)

e6

~f u 2:: 3e6 , If u < 3e6 ,

then v 2:: -'l/J (R) at all points in space and time. It is easy to see that the function 'IjJ hM all of the properties claimed in the statement of the corollary. 0

7.

WHEN BREATHERS AND SOLITONS ARE EINSTEIN

43

REMARK 1.69. A version of Corollary 1.68 for the Ricci tensor may be proved directly from the evolution of the curvature operator eigenvalues, using the maximum principle for systems. See [218] for details. COROLLARY 1.70 (Ricci flow on 3-manifolds: R bounds Rm). For any

solution (M3, 9 (t)) of the Ricci flow on a closed 3-manifold, if R is uniformly bounded, then IRml is also uniformly bounded. PROOF. If R :::; C, then 1/ ;::: -C' for some constant C'. If A ;::: J1 ;::: 1/ are the eigenvalues of Rm, then R = A + J1 + 1/ and A :::; R - 21/ :::; C + 2C'. 0

A nice application of the Hamilton-Ivey estimate is the following (see [218]). THEOREM 1.71 (Shrinking breathers on closed 3-manifolds are Einstein).

The only solitons (or breathers) for the normalized Ricci flow on a closed connected 3-manifold M are constant sectional curvature metrics. The example of Koiso's shrinking soliton (see Section 7 of Chapter 2) shows that this result cannot be extended to dimension 4. PROOF. By Proposition 1.66, either the metric is Einstein (which is equivalent to constant curvature in dimension 3) or Rmin > 0; in the latter case, the breather is a shrinking breather. By Corollary 1.68, for the unnormalized Ricci flow,

Rm > _ 1/J (R) id -7 0 as R -7 00 R R ' as t -7 T, where [0, T) is the maximal time interval of

and Rmin (t) -7 00 existence. So the sectional curvature becomes asymptotically nonnegative under the unnormalized flow. Since we assume our solution to be either a soliton or a breather, the curvature must have been nonnegative to begin with. In [179]' Hamilton has shown that either the sectional curvature becomes strictly positive immediately or M splits locally as a product of a l-dimensional flat factor and a surface with positive curvature, and this splitting is preserved by the flow. In the former case, we know 9 converges to a metric of constant positive sectional curvature under the normalized flow. To rule out the latter case, consider the evolution equation for r under the normalized flow:

!J

RdJ1= -

J(~~,RC-~R9)dJ1'

where the volume is fixed at one and the pointwise inner product is given by contraction using the metric. We can calculate the integrand on the right as

44

1. RICCI SOLITONS

Because of the flat factor, R2 = 21 Rc 12 , and therefore dr / dt = r 2/3. This implies that r increases without bound, which is impossible for a soliton or a breather. 0 A similar (but more elaborate) argument based on a pinching set can be used to prove that a nontrivial soliton for the normalized Ricci flow on a compact Kahler surface must have curvature at least as negative as Koiso's example; see [222].

8. Perelman's energy and entropy in relation to Ricci solitons The notion of gradient Ricci soliton has motivated the discovery of monotonicity formulas for the Ricci flow, which in turn have useful geometric applications. Here we consider some monotone integral quantities. In particular, in Chapter 5 we shall further study Perelman's energy functional: (1.68)

where (M n , g) is a closed Riemannian manifold and f : M ---+ lR. As we will see in (5.31) and (5.41), this functional is nondecreasing under the following set of evolution equations (see below for a motivation for considering (1.70)): (1.69) (1.70)

agij _ -2R·· at ~J' af 2 at = -ilf + IV' fl - R.

In particular, we have the following. THEOREM 1.72 (Energy monotonicity). For any solution (g (t) , f (t)) of (1.69)-(1.70) on a closed manifold M n , we have

:tF(g(t),J (t)) = 2 Hence -9t F (g(t) , f (t))

(1.71)

1M I~j + V' V'jfI2 e- f dp, ~ o. i

= 0 at some time to if and only if Rij + V'i V'jf == 0

at time to. That is, 9 (to) is a steady gradient Ricci soliton flowing along V' f (to). We can motivate the consideration of equations (1.69)-(1.70) by seeing how it relates to a steady soliton 9 flowing along a gradient vector field V' f and in canonical form. By (1.14) and (1.25),

~~ = IV' fl2 = IV' fl2 -

R - ilf,

which is (1.70). A similar consideration for shrinking gradient solitons can be used to motivate the study of Perelman's entropy functional W (g, f, T) discussed in

8.

PERELMAN'S ENERGY AND ENTROPY IN RELATION TO RICCI SOLITONS

Chapter 6 and the associated equations for g, f and r analogous to F we define in (6.1) the entropy:

W(g, f, r)

~ 1M

[r (R + IV' f12)

+f

-

45

> 0. In particular,

n] (47rr)-n/2 e-f dJ.L.

Under the system of equations

a

(1.72)

fJt gij

= -2Rij ,

(1.73)

af 2 n at=-f).f+lV'fl-R+2r,

(1.74)

dr =-1 dt '

we shall show in (6.17) the following. THEOREM 1.73 (Entropy monotonicity). If (g (t), f (t), r(t)), r (t) is a solution of (1.72)-(1.74) on a closed manifold Mn, then

> 0,

d

dt W(g(t), f(t), r(t)) =

1M 2r I~j + V'iV'jf -

;; 12 (47rr)-n/2 e- f dJ.L

~ 0.

ft

Note that the right-hand side (RHS) vanishes, i.e., W(g(t), f(t), r(t)) = 0, if and only if 9 (t) is a shrinking gradient Ricci soliton. The above monotonicity formulas beautifully display the utility of considering Ricci solitons. REMARK 1.74. Assuming that (M n , 9 (t)) is a shrinking gradient Ricci soliton in canonical form (1.13) with>' = -1, we have Rc (g (t)) where ~;

+ V'g(t)V'g(t) f

1

(t) - 2r g (t) = 0,

= -1, and hence (1.14) implies f satisfies (1.73).

Finally we consider the expander entropy of Feldman, Ilmanen, and one of the authors [143]. Define the functional W+ (g, f+, r)

~ 1M

Under the system (1. 75) (1.76) (1.77) we have the following.

[r (R + IV' f+12) - f+

+ n]

(47rr)-n/2 e- f +dJ.L'

1. RICCI SOLITONS

46

THEOREM 1.75 (Expander entropy monotonicity formula). For a solution (g (t), f+ (t), T(t)), T (t) > 0, of (1.75)-(1.77) on a closed manifold Mn,

d

dt W+(g(t), f +(t), T(t)) =

1M 2T I~j + '\h'Vjf+ + ;; 12 (47rT)-n/2 e- f +dj.L ~ O.

Here ftW+(g(t),J+(t),T(t)) = 0 if and only if g(t) is an expanding gradient Ricci soliton. Recall that from Proposition 1.7

8;j ~ Rij

(1.78)

+ \//vjf + 2ETgij ~ 0,

where T (t) ~ c:t+ 1 > 0 and ~ denotes an equality which holds for a gradient Ricci soliton (Mn,g(t),f(t),E) in canonical form. EXERCISE

1. 76. Show that

(1) (1.79)

(2) (1.80)

Moreover, if f has a critical point in space, then C (t) is independent of t.

9. Buscher duality transformation of warped product solitons In this section we describe an interesting duality transformation for solutions of the modified Ricci flow on certain warped products which in particular take gradient Ricci solitons to gradient Ricci solitons. The consideration of these warped products with tori of potentially infinite dimensions leads to Perelman's energy functional. 9.1. A metric duality transformation. Let (Mn,g) be a Riemannian manifold and let (Pq, h) be a flat manifold such as a torus or Euclidean space. Given a function A : M ---t (0,00), consider the warped product manifold (M, g) XA (P, h) which is the Riemannian manifold (M x P,g + Ah). The Buscher duality transformation (see [38]' [39]) takes the metric to the metric Ug~g+A-lh.

(This is a special case of T -d uality in string theory.) Although its definition is simple, the transformation has some surprising properties. Let {xi} ~=1

9. BUSCHER DUALITY TRANSFORMATION

47

and {ya}~=l be local coordinates on M and 'P, respectively, such that ha{3 ~

h

(a~o., ~ ) = fJa{3.

We then have

P9ij = 9ij, P9a(3

=

AfJa{3,

and the rest of the components are zero, where p

.p(8 8) 9 8xi '8xj ,

P

9ij =;=

.p(8 8) 9 8ya' 8y{3 ,

9a{3 =;=

and similarly for ~ 9. The explicit formulas below are from Haagensen [175] (see also (1.38) or §J in Chapter 9 of Besse [27] for curvature formulas for warped products). LEMMA

1.77. The Christoffel symbols of p9 are pr~. ~J

= r~.

~J'

p {3 _ {31 . ria - fJa2V'~logA,

A

.

=

prij

p .

= -fJa {3 '2 V'~ log A,

pr~{3

=

r~{3

pr~j

= 0,

and likewise, the Christoffel symbols of the dual metric ~9 are

~rfj

= rfj ,

~r{3 = -fJ{3a ~2 V' . log A , ~a

~

. r~{3

~

1·

= fJ a {3 2A V'~ log A, ~rl'{3 a = ~riaJ. = ~r~. = 0 • ~J

LEMMA

1.78. The Ricci tensor of p9 is 9iven by

~ [~logA+ ~ lV'logAI 2] fJa{3,

PRa{3

=-

PRai

= 0,

P~j

= Rij - ~ V'i V' j log A - ~ V'i log A V' j log A,

and the Ricci tensor of ~9 is

~ Ra{3 = - 2~ ( -~ log A + ~ IV' log A12) fJa{3, ~Raz. -- 0 , M

~j

M

= ~j +

q q 2V'iV'j log A - '4V'i log AV'j log A.

48

1.

RICCI SOLITONS

LEMMA 1.79. The scalar curvature of P9 and ~ 9 are given by PR =

R -

q~logA _

~ R = R + q~ log A EXERCISE 1.80. Show that if A P

Rij =

~j

q (q: 1) IVlogA1 2 ,

- q (q 4+ 1) IV log AI2 .

~ exp (-~f) (i.e., log A

+ ViVjf -

1 q

q + 1 IV f1 2 , q

~ R = R - 2~f -

q + 1 IV fl2 . q

R

-~f), then

-Vi/Vjf,

+ 2~f -

PR =

=

Hence lim R P = R

q-+oo

+ 2~f -

IV fl2 ,

are Perelman's modified Ricci tensor and scalar curvature (see also (5.18) and (5.19)). As a consequence of the above exercise, if h has unit volume, we then have

lMxP Reg) dJ.L. g = 1M (R + 2~f - q; 1 1v f12) e- f dJ.Lg = 1M (R + q ~ 1 IV fl2 ) e- f dJ.Lg.

Taking q ~

00,

this limits to F (g, f) defined in (1.68).

9.2. Buscher duality. For a warped product solution of the modified Ricci flow of the above type we have the following. THEOREM 1.81 (Buscher duality). If Pg

(t) = 9 (t)

+ exp (-~ f q

t

E

(t))

t

dyO< Q9 dyO<,

0<=1

I, satisfy the modified Ricci flow

(1.81)

:t

Pgab = -2

e

Rab + 2 PVa PVb P
where P

~ 9 (t) + exp

Gf (t)) t,

dyo 0 dyo

9. BUSCHER DUALITY TRANSFORMATION

49

satisfy (1.82) where

Uc/> ~ ~ c/> + J. REMARK

1.82. Note that ~V o:f = Uv o:f =

o.

The functions 11 c/> and Uc/> are called dilatons and the transformation from 11 c/> to Uc/> is called the dilaton shift. The Buscher duality transformation takes solutions of the modified Ricci flow with dilaton ~ c/> to solutions of the modified Ricci flow with dilaton Uc/>. LEMMA

1.83 (Buscher duality preserves modified Ricci tensor after dila-

ton shift).

URo:.8 + 2 Uv 0: ttV.8 Uc/> = URo:i + 2 Uv 0: "Vi Uc/>

=

~2

( 11 Ro:.8 + 2 I1V 0: I1V.8 11 c/> ) ,

~ Ro:i + 2 I1V 0: I1Vi 11 c/> = 0,

URij+2 UVi "Vj Uc/>= l1~j+2 I1Vi I1Vj 11c/>. REMARK

1.84. If log A = - ~ f, then

~ Rij + 2 ~Vi

I1Vj 11c/> = Rij + ViVjf -

~VdVjf + 2V iVj ~c/>, q

1 q

tt Rij + 2 UVi UVj Uc/> = Rij - ViVjf - - VdVjf + 2V i Vj Uc/>. Suppose limit as q ~

%t 00,

~ gab = -2 11 Rab so that ~ c/> = O. By Exercise 1.80, taking the the equation for the metric gij (t) on the base manifold is

a

atgij

= -2 (Rij + ViVjf).

Note that d/-l (11 g) = e- f d/-l/\ dyl /\ ... /\ dyq. The effective action of (11 g,11 c/» is (e.g., see Alvarez and Kubyshin [2], equation (26))

iMXP (Reg) + 41v 11c/>12) e- 2b4>d/-l eg) = iM (R + 2b.f - q: 1 IV fl2 + 41v ~ c/>12) e- 2b4>e- f d/-l,

S eg,11 c/» =

and similarly for the dual pair (Ug,U c/» • (Note that by taking c/>~ = 0, we have S (l1g, 0) = fMxp R (l1g) d/-lb g.) Buscher duality preserves the effective action:

50

1. RICCI SOLITONS

9.3. Examples. The Buscher dual of a rotationally symmetric soliton solution on a surface is the soliton For example,

dr 2 + tanh 2 r d0 2

dr 2 + coth 2 r d0 2

dr 2 + tan2 r d0 2

dr 2 + cot 2 r d0 2

dr 2 + r 2d0 2

dr 2 + r- 2d0 2

dr 2 + d0 2

dr 2 + d0 2

Let (Nm, h) be an Einstein metric with Rc = £g, c E R Consider the doubly-warped product metric on R2 X Nm: ~g ~ ds 2 + F (s)2 d0 2

-+ G (s)2 h,

where s is the radial coordinate, 0 is the coordinate on the circle, and F and G are positive functions. The gradient Ricci soliton equation ~ Rab + 2 ~Va ~VbI = 0 becomes (see equation (1) in [219]) the following system for the triple (F, G, J): (1.83)

G"

F"

-21" = -C - F'

(1.84)

F" = -n F'G' - l' F',

(1.85)

C =

G

G"

c G2 - (n - 1)

(G')2 G

-

F'G' G' FG - l' G .

The Buscher dual metric is 1 ~g = ds 2 + --2d02

F(s)

+ G (s)2 h.

We know that ~g is a steady gradient Ricci soliton if ~ 9 is, and so we leave it to the reader to verify that indeed the triple (*, G, 1 - log F) is a solution to (1.83)-(1.85) if (F, G, J) is. 10. Summary of results and open problems on Ricci solitons

In this section we collect some known results and open problems about the properties and classification of gradient Ricci solitons. Besides the discussion earlier in this chapter, one may consult [111] for some of the proofs.

10.

SUMMARY

OF

RESULTS AND OPEN PROBLEMS ON RICCI SOLITONS

51

10.1. Gradient Ricci solitons on surfaces. The following is a compendium of known results about complete 2-dimensional gradient Ricci solitons with bounded curvature (shrinkers, steadies, and expanders). In proving them, one may use the fact that if (M 2 ,g) admits a nontrivial Killing vector field X which vanishes at some point 0 E E, then (M,g) is rotationally symmetric. (1) A shrinker has constant positive curvature. In particular, the underlying surface is compact. (2) A steady is either flat or the cigar. (3) A compact expander has constant negative curvature (if X (M) < 0, then there are no nonzero conformal Killing vector fields). (4) An expander with positive curvature is rotationally symmetric and unique up to homothety (see [241] and [111]). In part (4), one can apply the arguments of subsection 3.1 of this chapter and the following result (see [286] and [111]). THEOREM 1.85. If (Mn,g) is a gradient Ricci soliton on a noncompact manifold with Rij ~ ERgij for some E > 0, where R ~ 0, then R decays exponentially in distance to a fixed origin. In particular, if (M2, g) is an expanding gradient Ricci soliton on a surface, then R decays exponentially. A complete proof, using a more direct method, is given in [241]. PROBLEM 1.86. Are there any other expanders on a surface diffeomorphic to ]R2 besides the positively curved rotationally symmetric expander and the hyperbolic disk? PROBLEM 1.87. Do all complete 2-dimensional gradient Ricci solitons have bounded curvature? Are all complete 2-dimensional Ricci solitons gradient? Note that in dimension 3 there are complete homogeneous expanding solitons which are not gradient; see Section 5 of this chapter.

10.2. Gradient Ricci solitons on 3-manifolds. In dimension 3 we have the following results. This subsection is abbreviated since we pose some more problems in the next subsection for dimensions at least 3. (1) (Perelman) Any nonflat shrinker with bounded nonnegative sectional curvature is isometric to either a quotient of the 3-sphere or a quotient of S2 x R In particular, any shrinker with bounded positive sectional curvature is isometric to a shrinking solution with constant positive sectional curvature. (2) There exists a rotationally symmetric steady with positive sectional curvature, namely the Bryant soliton. (3) There exists a rotationally symmetric expander with positive sectional curvature.

52

1.

RICCI SOLITONS

PROBLEM 1.88. Are there any 3-dimensional steady gradient solitons besides a flat solution, the Bryant soliton, and a quotient of the product of the cigar and 1R? This is equivalent to asking if a steady gradient Ricci soliton with n = 3 and sect (g (t)) > 0 is isometric to a Bryant soliton.

10.3. Gradient Ricci solitons in higher dimensions. Here we assume n ~ 3. Note that any expanding or steady Ricci soliton on a closed n-dimensional manifold is Einstein. PROBLEM 1.89. Is a shrinking gradient Ricci soliton with n Rm (g (t)) > 0 compact?

~

4 and

PROBLEM 1.90. Is a compact shrinking gradient Ricci soliton, n ~ 4, and Rm (g (t)) > 0 isometric to a shrinking constant positive sectional curvature solution?16 PROBLEM 1.91. Are there n-dimensional steadies with positive curvature operator besides the Bryant soliton? PROBLEM 1.92. Are there any n-dimensional expanders with positive curvature operator besides the rotationally symmetric one? PROBLEM 1.93. Does there exist an expander with n ~ 3 and positively pinched Ricci curvature, i.e., Rij ~ ERgij for some E > 0, where R> O? By Theorem 1.85, such an expander has R decaying exponentially.

11. Notes and commentary Section 1. The first occurrence of the notion of a Ricci soliton in the literature is in Friedan [145], where non-Einstein Ricci solitons are called quasi-Einstein metrics. See Besse [27] for a comprehensive treatment of Einstein manifolds. The Gaussian soliton first appeared in §2.1 of Perelman [297]. In the case of surfaces we have encountered Ricci solitons in Chapter 5 of Volume One. There, solitons motivated several aspects of Hamilton's original proof of convergence of the Ricci flow for surfaces with R (go) > 0, including the scalar curvature, Harnack, and entropy estimates, as well as the estimate which shows the metric approaches a soliton. See in Volume One, Corollary 5.17 on p. 115, Proposition 5.57 on p. 145, Proposition 5.39 on p. 134, and Corollary 5.35 on p. 130. Expanding solitons are of interest because they constitute borderline cases for Hamilton's Harnack inequality. (Expanding solitons also model the formation of certain Type III singularities.) In fact, as we mentioned in Section 2 of this chapter and will discuss in more detail in Part II, the consideration of quantities which vanish on solitons led to the discovery of the Harnack quadratic. Steady solitons, where the isometry class of the metric is independent of t E (-00,00), occur as limits of Type II singularities. 16T he answer to this problem is 'yes' and follows from the recent work of B6hm and Wilking [30] (see Part II of this volume).

11.

NOTES AND COMMENTARY

53

Section 4. Einstein metrics are special cases of Ricci solitons. In general, if a warped product metric 9 is Einstein, then the fiber ('P, g) is Einstein. If the base is I-dimensional, then provided w (r) is not constant, the metric 9 = dr 2 + w (r)2 9 satisfies Rc (g) = )..g if and only if w' (r)2

+ ~w (r)2 =

_P_, n n-I where Rc (9) = pg and n = dim 'P. Without loss of generality (e.g., up to scaling and diffeomorphism), we have the following cases:

I

p =

~~i I).. = R~) I w (r) I

9

- (n - 1)

-n

0

-n

eT

including hyperbolic cusp

n-I

-n

sinhr

including hyperbolic space

n-I

0

r

Ricci flat cone

n-I

n

smr

including sphere

0

0

1

Ricci flat product

coshr including hyperbolic with two ends

As a consequence, we have the following (see Theorem 9.110 on p. 268 of [27]). THEOREM 1.94 (Einstein warped products over I-dimensional base). If a warped product (M, g) over a I-dimensional base, with the dimension of the fiber at least 2, is a complete Einstein manifold, then either (1) 9 is a Ricci flat product, (2) M is topologically a cone on P and the fiber is Einstein with positive scalar curvature, or (3) the base is JR, the fiber is Einstein with nonpositive scalar curvature, and 9 has negative scalar curvature.

CHAPTER 2

Kahler-Ricci Flow and Kahler-Ricci Solitons Symmetry, as wide or narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty, and perfection. - Hermann Weyl

The Kahler-Ricci flow is simply an abbreviation for the Ricci flow on Kahler manifolds. In this chapter we first review some basic definitions and properties for Kahler manifolds and survey some of the fundamental results on the existence of Kahler-Einstein metrics. Then we discuss elementary properties of the Kahler-Ricci flow and state some of the fundamental longtime existence and convergence results. We also give a survey of KahlerRicci solitons. Some other highlights of this chapter are an exposition of tensor calculations in holomorphic coordinates, the proof of the long-time existence and convergence of the Kahler-Ricci flow on Kahler manifolds with Cl < 0, construction of the Koiso solitons and other U(n)-invariant solitons, proofs of differential Harnack estimates under the assumption of nonnegative holomorphic bisectional curvature, and a survey of uniformization-type results for complete noncompact Kahler manifolds with positive bisectional curvature. We assume the reader either has some knowledge of Kahler geometry or will read other references on Kahler geometry along with this chapter, so we make no attempt to be completely self-contained. l The latter chapters do not depend on this chapter and the reader interested only in Riemannian Ricci flow may skip this chapter. 1. Introduction to Kahler manifolds

In this section we introduce the basic concepts of complex and Kahler manifolds including the point of view of having an almost complex structure on an even-dimensional Riemannian manifold satisfying natural properties. Let M be a real 2n-dimensional differentiable manifold. A system of holomorphic coordinates on M is a collection {Zi : Ui ---t Zi (Ui) c en} , where {Ui} is a cover of M and Zi are homeomorphisms such that the maps

Zi

0

zjl : Zj (Ui n Uj ) ---t Zi (Ui n Uj )

lSee the notes and commentary at the end of this chapter for some references on complex manifolds and Kahler geometry. 55

56

2. KAHLER-RICCI FLOW

are holomorphic (complex analytic), and hence biholomorphic, whenever

Ui n Uj i= 0. Two systems of holomorphic coordinates {Ui' Zi} and {Vj, W j} on M are equivalent if whenever Ui n Vj i= 0, Wj 0 Zi- 1 :

Zi (Ui

n Vj)

----t

Wj

(Ui n Vj)

is a biholomorphism. A complex structure on a differentiable manifold M is an equivalence class of systems of holomorphic coordinates. A complex manifold is simply a differentiable manifold with a complex structure. A complex manifold has a natural real analytic structure and is orientable (using holomorphic coordinates {zO: ~ xO: + Ryo:} an orientation is

:=1 '

{/xr, i!;r, ... ,

determined by requiring that the frame a~n , a~n } be positively oriented). In this chapter and only in this chapter we shall use Mn to denote a complex manifold M of complex dimension n (half of the real . ) ,.I.e., n:::;=. d'Ime MId' · d ImenSlon = 2' Im]R M . A real sub manifold N of a complex manifold M n is a complex submanifold if for every pEN there exist holomorphic coordinates {ZO:}:= 1 in a neighborhood U of p such that

N

nU = {q E U : zk+

1

(q) = ... = zn (q) =

o} .

Clearly a complex submanifold is itself a complex manifold. In the above definition, k is the complex dimension of N. Some elementary examples of complex manifolds are (1) complex Euclidean space C n , (2) complex projective space cpn, (3) complex submanifold of cpm = algebraic manifold = zero set of a finite number of homogeneous polynomials, (4) complex torus C n Ir, where r is a lattice. A complex structure defines at each point p E M a map J : T Mp ----t T Mp by J = d (z-l 0 R 0 z) , where z are holomorphic coordinates defined in a neighborhood of p (this definition is independent of the choice of z). Since R 0 R = - iden, it is easy to see J2 = - idTM . In general, given an even-dimensional differentiable manifold M, an automorphism J : T M ----t T M is called an almost complex structure if J2 = - idTM . (A manifold with an almost complex structure is called an almost complex manifold.) Hence a complex structure induces an almost complex structure. We say that an almost complex structure is integrable if there exists a complex structure which induces the almost complex structure. Given an almost complex manifold (M, J), the Nijenhuis tensor is defined by

N J (X, Y) ~ [JX, JYJ - J [JX, YJ - J [X, JYJ - [X, YJ for X, YET M. By the Newlander-Nirenberg Theorem, a necessary and sufficient condition that an almost complex structure be integrable is that the Nijenhuis tensor vanish.

1. INTRODUCTION TO KAHLER MANIFOLDS

57

It is interesting that 8 6 admits an almost complex structure. In fact 8 2 and 8 6 are the only even-dimensional spheres which admit almost complex structures (see §8 of Chern [95] for example). The following is a longstanding unsolved question. 2.1 (Existence of complex structure on 8 6 ). Does 8 6 admit a complex structure? More generally, one may ask which closed evendimensional manifolds admit almost complex structures but not complex structures. PROBLEM

Let (M, J) be an almost complex manifold. A vector field V is an infinitesimal automorphism of the almost complex structure if the Lie derivative of J with respect to V is zero, i.e., (2.1)

LVJ

= O.

Note that (LVJ) (W)

= LV (JW) - J (LVW) = [V, JW]- J ([V, W]),

so (2.1) is equivalent to J ([V, W]) = [V, JW] for any vector field W. There are various equivalent ways to define a Kahler manifold. We say that a Riemannian manifold (M,g) with an almost complex structure J : T M ~ T M is a Kahler manifold if the metric 9 is J-invariant (or Hermitian): 9 (JX, JY) = 9 (X, Y)

and J is parallel: \7 J = 0

or equivalently, \7 x (JY) = J (\7 x Y)

for all X, Y, where \7 is the Riemannian covariant derivative. 2 The metric 9 is called a Kahler metric. Note that almost complex structures which yield Kahler manifolds are necessarily integrable. Indeed, by the Newlander-Nirenberg Theorem, we only need to check that the Nijenhuis tensor vanishes for a Kahler manifold: NJ (X, Y)

= \7 JxJY -

\7 JyJX - J (\7 JXY - \7yJX)

- J (\7 x JY - \7 JY X) - \7 x Y

=-

[J (\7 JX Y) - \7 JxJY]

+ [J(\7yJX) -

+ \7 Y X

+ [J (\7 JY X)

- \7 JyJ X]

\7 y J(JX)]- [J(\7xJY) - \7xJ(JY)]

=0.

A complex manifold (M, J) is called a Kahler manifold if it admits a Kahler metric. 2Since 0

= (V xJ) (Y) = V x

(JY) - J (V x Y).

58

2. KAHLER-RICCI FLOW

Given an almost complex manifold (M, J), the complexified tangent bundle is TeM ~ T M ®lR C. For each p E M we may extend the almost complex structure J to a complex linear map

Jc : TeMp

~

TeMp.

Since (Jc)2 = - idTcM p ' the eigenvalues of Jc are Rand define the holomorphic tangent bundle by

Tl,o M ~ {V

E

R.

We

TeM : Je (V) = HV}

and the anti-holomorphic tangent bundle by

TO,l M ~ {V E TeM : Je (V) = -HV} . This gives us a decomposition

TeM = Tl,o M EB TO,l M. A vector in Tl,o M is called type (1,0) and a vector in TO,l M is called type (0,1) . Alternatively, we can obtain the above decomposition by decomposing a (real) tangent vector VET M as V = Vl,o + VO,l, where

Vl,o ~

'12 (V -

HJV)

E

Tl,o M,

VO,l ~

'12 (V + HJV)

E

TO,l M.

I.e.,

Tl,oM = {Vl,o: V

E

TM},

TO,lM = {VO,l : V

E

TM}.

Since TeM is the complexification of a real vector space, the complex conjugate (bar operation), X + A Y ~ X - AY, where X, YET M, is defined on TeM. Given Z E TeM, the real part of Z is defined by Re (Z)

~ ~ (Z + 2) .

Note that if VET M, then 2 Re (Vl,O)

= 2 Re (VO,l) = V.

The almost complex structure satisfies

Je (if) = Je (V),

V

E

TeM,

and we have Tl,oM = TO,lM and TO,lM = Tl,oM. The complexified cotangent bundle is TcM = (TeM)* = T* M ®lR C, which decomposes into

Tc M = Al,oM EBAO,lM, where Al,oM ~ (Tl,oM)* and AO,lM ~ (TO,lM)*.

1. INTRODUCTION TO KAHLER MANIFOLDS

59

A covariant tensor of type (p, q) is a section of ®p,q M ~ (®P A1,0 M) ® (®q AO,lM).3 The complex conjugate extends to the tensor bundles and

Q9p,qM=Q9q,PM. We say that a differential (p + q)-form "1 is of type (p, q) if it is a section of the vector bundle AP,qM ~ (APT1,OM) A (AqTO,lM) c Ap+qTcM. We denote the space of (p, q)-forms by Op,q (M). We say that a (p,p)-tensor (or form) "1 is real if ij = "1. When we are considering complex manifolds, it is most natural to carry out calculations in holomorphic coordinates. Let {zC<} be local holomorphic coordinates. We may write zC< ~ xC< + Ayc<, where xC< and yC< are realvalued functions. Define dzC< ~ dxC<

+ HdyC< ,

diC< ~ dxC< - HdyC< and

8~C< ~ ~ (8~C< - H 8~C< ~ ~ (8~C< + H

8:C<) , 8:C<) ,

so that dzC<

(8~f3 )

=

6$,

diC<

(8~f3 )

=

6$,

(8~f3 ) = diC< (8~f3 ) = o. -JJa and ib = b· The holomorphic

dzC< Note that

-£a

=

tangent bundle

Tl,o M is locally the span of the vectors {8~"'} :=1 . Given two overlapping local holomorphic coordinates {zc<} and {wf3} ,the transition matrix relating 8 d 8 . {8z"'}n h· h t· fi 8z'" an 8w/3 IS 8w/3 c<,f3=1' w lC sa IS es 8 8wf3 =

n

L

c<=l

8 zC< 8 8wf3 . 8 zC<·

The anti-holomorphic tangent bundle yo,l M is locally the span of

{ib} ;=1.

NOTATION 2.2. Henceforth we shall use the Einstein summation convention where each pair of repeated indices consisting of an upper index and

a lower index is summed from 1 to n (sometimes, as a reminder, we include the summation symbol, however we always sum over repeated indices unless otherwise indicated). 3Not to be confused with a (P, q)-tensor in Riemannian geometry which is a section of (®P T* M) ® (®q TM).

2. KAHLER-RICCI FLOW

60

EXERCISE

2.3. Given a (p, q)-form

. ~ ~ dZ i1 ./ - n. 'nl"'tp)l"')q

n -

1\

•••

1\

dz ip

1\

1\

1\

1\

dZ}l

1\

1\

•••

1\

1\

dz}q ,

show that its complex conjugate is given in local coordinates by ii == (-I)pq n. ,. . dz iI A·,· A dz jq A dZ'il A··· A dz'ip E nq,p (M) . ./ . 'nl'''tp)l''')q The exterior differentiation operator d maps Ap,qM into AP+1,qM EB Ap,q+l M. Corresponding to this decomposition of the image of d, we have

d~8

+ a,

where

8: AP,qM

~

Ap+1,QM,

a: Ap,q M

~

AP,Q+1 M.

We extend the Riemannian metric 9 complex linearly to define ge : TeMp x TeMp

~

C.

Similarly 'Ve : TeM x Coo (TeM)

~

Coo (TeM) is the complex linear extension of 'V : T M x Coo (T M) ~ Coo (T M) with the convention that 'VxY ~ 'V (X, Y) and ('Veh Y ~ 'Ve (X, Y). The complex linear extension of Rm is denoted by Rme . Let

Since ge (V, W) = ge Hermitian condition:

(V, W)

= ge

(W, V) , these coefficients satisfy the

Similarly, we define

90.(3

~ ge (8~0.' 8~(3 )

,

We claim 90.(3 = 90i.i3 = O. Indeed, if X, Y E T1,0 M, then ge (X, Y)

= ge (JX, JY) = ge (yCIx, yCIy) = -ge (X, Y),

which implies ge (X, Y) = O. Similarly, if X, Y E TO,l M, then we also have ge (X, Y) = O. Thus, in local holomorphic coordinates, the Kahler metric takes the form ge = 90.i3 ( dzo. 0 dz(3 + dz(3 0 dZo.) . EXERCISE 2.4. Show that if a a is real if and only if

=

Aao.i3dzo. A dz(3 is a (1, I)-form, then

1.

INTRODUCTION TO KAHLER MANIFOLDS

61

Given a J-invariant symmetric 2-tensor b, we define a J-invariant 2-form ,8 by ,8(X,Y)~b(JX,Y).

(We check that ,8 (Y, X) = b (JY, X) = b (-Y, JX) = -,8 (X, Y).) We call,8 the associated 2-form to the symmetric 2-tensor b. The Kahler form w, on a Riemannian manifold (M, J,9) with an almost complex structure and whose metric is Hermitian, is defined to be the 2-form associated to 9: w (X, Y)

~ 9

(JX, Y),

which is a real (1, I)-form. EXERCISE 2.5. Show that if 9 is a Kahler metric, then w is a closed 2-form. In fact, w is parallel. HINT: See [27], Proposition 2.29 on p. 70. SOLUTION: We compute (\7 xw) (Y, Z) = X (w (Y, Z)) - w (\7 x Y, Z) - w (Y, \7 x Z)

= X (9 (JY, Z)) - 9 (J (\7 x Y) , Z) - 9 (JY, \7 x Z) = (\7 x9) (JY, Z) = 0, where we used the definition of w, J (\7 x Y) = \7 x (JY) , and 9 is parallel. On a Kahler manifold (M, J,9), in local holomorphic coordinates, the Kahler form is n

W

=

R

L

9o.fjdzo. 1\ d-z{3.

o.,{3=1

b)

REMARK 2.6. In our convention, dzo. I\d-z{3 (-/z-r, = !8~8r Note that for the standard Euclidean dx 2 + dy2 metric on en, we have 9o.fj = !8o.{3 and 9c = (dzo. 0 d-zo. + d-zo. 0 dzo.) .

! r::=1

Since 9 is Kahler, we have the Kahler identities:

G

(2.2)

G

Gz"Y 9o.fj = Gzo. 9"Yfj,

which are equivalent to dw

[w]

= O. The real cohomology class

E H1,1(M;~) ~ H2(M;~)

is called the Kahler class of w. EXERCISE 2.7 (Characterization of Kahler condition). If (M, J,9) is a triple consisting of a Riemannian manifold, an almost complex structure, and a Hermitian metric, then dw = 0 if and only if \7 J = O. That is, 9 is Kahler if and only if the real (1, I)-form w is closed.

2. KAHLER-RICCI FLOW

62

An oriented Riemannian surface (M,9) has a natural complex structure and 9 is a Kahler metric with respect to this complex structure. In particular, define the almost complex structure J : T M --t T M as counterclockwise rotation by 90° with respect to the orientation and metric (clearly J2 = - idT M)' Since dimlR M = 2, we have dw = 0, which implies \1 J = O.

2. Connection, curvature, and covariant differentiation In this section we consider the connection, curvature, and covariant differentiation on a Kahler manifold (M n , J,9). Our emphasis, in the style of the book by Morrow and Kodaira [275], is to calculate geometric quantities in local holomorphic coordinates, where the formulas are particularly elegant and simpler than their Riemannian counterparts. These calculations shall prove useful in our study of the Kahler-Ricci flow. The Christoffel symbols of the Levi-Civita connection, defined by

8 ~~('Y 8 ( ) \1 IC o~'" 8z/3 -;- L...J r Ot/3 8z-Y 'Y=1

(\1 c)

0

-

Y 8) + r 'Ot/3 8z'Y '

n(r'Y --8 + r'Y-8) --

8 == ~

oz'" 8z/3 . L...J -y=1

Ot/3 8z'Y

Ot/3 8z'Y

'

etc., are zero unless all the indices are unbarred or all the indices are barred. To see this, from the Riemannian formula for the Christoffel symbols and since we are complex linearly extending the covariant derivative, we have the following using (2.2). 2.8 (Kahler Christoffel symbols). Let 9 Ot (3 be defined by 9'Y(39 Ot (3 Then, in holomorphic coordinates, we have

LEMMA 8~.

(

2.3)

-y

=

_ 1 -y6 (8 _ 8 _ 8 ) _ 'Y6 8 _ 8zOt9/3li + 8z/39Otli - 8zli9Ot/3 - 9 f)zOt9/3li

r Ot/3 - '2 9

and

Y -- r'Y/3Ot r 'Ot/3 (in fact, the last equality is equivalent to the Kahler condition).4 Similarly

r~(3 = ~ 9 -y6 (f)~Ot 9(36 + f)~/3 9Ot6 - 8~li 9Ot(3) r~(3

= 0,

= 0, and so forth. EXERCISE

2.9. Show that Y r '(i(3

-

-

r'YOt/3'

4We leave this as an exercise or see for example Theorem 5.1 in [275].

2. CONNECTION, CURVATURE, AND COVARIANT DIFFERENTIATION

63

SOLUTION. We compute 1

_ 1 18 (0

r aiJ - 29

_ {jiQ9f38

0

+ ozf3 9M -

0 _) _ 18 0 _ oz89af3 - 9 oz0l.9f38

- 0 = 9'Y8_ - = r'Y ozOl. 9f38 0I.f3. Similarly to ge and 'Ve, we may define Rme and Rce as the complex multilinear extensions of Rm and Rc. The components of the curvature (3, 1)-tensor Rme are defined by 00)0. 8 0 Rm e ( ozOl. 'ozf3 OZ'Y =:= ROI.iJ'Y Oz8

+ ROI.iJ'Y OziS'

00)0. 8 0 Rm e ( OzOI. 'ozf3 oz1 =:= ROI.iJ1 Oz8

+ ROI.iJ1 OziS'

is

0

is

0

etc. As a (4,0)-tensor the components of Rme, which are defined by ROI.iJ'Y iS

~ Rme (O~OI.' 0~f3' O~'Y' 0~8 )

,

satisfy etc. The only nonvanishing components of the curvature (3, 1)-tensor are

R~iJ'Y ' R~iJ1' R~f3'Y ' R~f3i . In particular, R~f3'Y = R~f3i = 0, etc. Hence, the only nonvanishing components of the curvature (4, O)-tensor are ROI.iJ'YiS' ROI.iJ18 , Raf3'YiS' Raf3i8.

Since a~o< r~'Y = 0, we have

(2.4) and thus we have the following lemma. 2.10 (Kahler Rm). In holomorphic coordinates, the components of Rme are 9iven by LEMMA

(2.5)

We have the identities ROI.iJ'YiS

= R'YiJOI.iS = ROI.iS'YiJ = R'YiSOI.iJ

and ROI.iJ'Y iS = Rf3a81· The vanishing of some of the components of the curvature (4, O)-tensor is related to the following.

64

KAHLER-RICCI FLOW

2.

EXERCISE 2.11 (J-invariance of curvature). Show that for a Kahler manifold Rm (X, Y) is J-invariant, i.e., Rm (X, Y) J Z = J (Rm (X, Y) Z) . SOLUTION. We compute Rm (X, Y) JZ = \lx\lyJZ - \ly\lxJZ - \l[X,y]JZ = \lx (J\lyZ) - \ly (J\lxZ) -

J (\l[X,y]Z)

= J (\lx\lyZ) - J (\ly\lxZ) - J (\l[X,y]Z) =

J (Rm (X, Y) Z) .

The components of Rcc are defined similarly:

(8~a' 8~/3 ) Rc c (8~a' 8~/3 )

Rc C

=;= Rai3' =;= R a /3,

etc. Tracing (2.4), we see that components of the Ricci tensor Rai3 = R~i3a are given by LEMMA 2.12 (Kahler Rc).

(2.6)

82

Rai3 = - 8z a8z/3 log det (9",s) .

It is easy to see that Rai3 = R/3o. and Ra/3 = RQi3 = O. EXERCISE 2.13. Show that for a Kahler manifold Rc is J-invariant, i.e., Rc (JX, JY) = Rc (X, Y). The Ricci form p is the 2-form associated to Rc, p (X, Y) =;=

1

2 Rc (JX, Y),

which is also a real (1, I)-form. EXERCISE 2.14. Show that if 9 is a Kahler metric, then the Ricci form p is a closed 2-form.

HINT: See Proposition 2.47 on p. 74 of [27]. The real de Rham cohomology class [2~P] =;= Cl (M) is the first Chern class of M. It is a beautiful fact that [2~P] only depends on the complex structure of M. The Ricci form, which is a real (1, I)-form, is in holomorphic coordinates: p = RRai3dza 1\ dz/3.

From (2.10) below and r~/3

= r~a 8

we have

8

8 za RWr = 8 z/3 Ra;'fl

2. CONNECTION, CURVATURE, AND COVARIANT DIFFERENTIATION

which is equivalent to the Ricci form being closed: dp (2.6) as p = -v'-Iaalogdet (9['8) . The complex scalar curvature is defined to be

= o.

65

We may express

R -;- 9 o13R0(3· ...!...

The complex scalar curvature is one-half of the Riemannian scalar curvature. (Exercise: Prove this.) 2.15. In this chapter, R always denotes the complex scalar curvature whereas in the rest of the book, R always denotes the Riemannian scalar curvature. In this chapter we shall usually refer to the complex scalar curvature simply as the scalar curvature. NOTATION

Holomorphic coordinates {ZO} are said to be unitary at a point p if 9 (a~Ct' (p) = 80 (3. That is, {a~a (p)} is a unitary frame for T 1,o Mp. In such a frame we have at p,

:=1

ib)

n

(2.7)

Ro13 =

L Ro/388

8=1

and 0=1

Note that for the standard Euclidean metric 1 n ge = 2 (dw O 0 diiP + diiP 0 dw O )

L

0=1

on

en the new coordinates {zo ~ ~Wo} :=1 are unitary.

{a},

1

Given Z E T 1,oM let X ~ Re(Z) = (z + z) E TM. The holomorphic sectional curvature in the direction Z is defined to be

(2.8)

K (Z)=Rm(X,JX,JX,X)

IXI 4

e.

Rm(Z+Z,J=I(Z - Z) ,J=I(Z - Z) ,Z+Z)

41Z1 4 Rme

(Z, Z, Z, Z)

IZI 4 Hence Ke (Z) = Rme (V, V, V, V) , where V ~ Zj IZI. In particular, if {ZO} is unitary at p, then Ke (-£0) = RO{io{i. We say that the holomorphic sectional curvature is positive (respectively, nonnegative) if Ke (Z) > 0 (respectively, Ke (Z)

~ 0) for all Z

{a}.

T 1,o M By (2.8), positive (respectively, nonnegative) Riemannian sectional curvature implies positive (respectively, nonnegative) holomorphic sectional curvature. Given Z,W E T 1,oM let X ~ Re(Z) and U ~ Re(W). The (holomorphic) bisectional curvature in the directions (Z, W) is defined

{a},

E

66

2. KAHLER-RICCI FLOW

to be J(,

(Z W) == Rm(X, JX,JU,U) = Rme (Z,Z, W, w)

e,

.

IZI 21WI 2

IXI21U12

Clearly we have Ke (Z, Z) = Ke (Z). We say that the bisectional curvature is positive (respectively, nonnegative) if Ke (Z, W) > 0 (respectively, Ke (Z, W) 2: 0) for all Z, WE Tl,O M Clearly positive (respectively, nonnegative) bisectional sectional curvature implies positive (respectively, nonnegative) holomorphic sectional curvature. We also have the following (see Proposition 1 on p. 32 of [270] for example).

{o} .

LEMMA 2.16. Po.sitive (respectively, nonnegative) Riemannian sectional curvature. implies positive (respectively, nonnegative) bisectional sectional curvature.

Similarly to the notation for the components of the complex Riemann curvature tensor, we define

V o.R{3"y8TJ

~ ((Vc) {}~'" Rm e) (0~{3' 0~'Y ' 0~8 ' O~TJ )

.

The second Bianchi identity says that

(2.9) since - ) + ( V {} Rme )(0 - 0 - 0 - 0) ( V {} Rme )( - 0 - 0 - 0 0 {}zO oz{3' oi'Y' oz8' oiTJ {}z'Y ozo.' oz{3' oz8' oiTJ

+ (V a;P {} Rme) (o~z'Y , -00 zo. , 00ZU

0

o~zTJ )

=0

(Le., V o.R{3"y8ii + V (3R;yo.8ii + V"yRo.{38ii = 0, where V"yRo.{38ii = 0). Taking the trace, we have

(2.10) Note that Vo.R{3;y = ~R{3;y - r~{3R8"y' The volume form is d/1g ~ ~wn, where w n ~ w 1\ n = dime M, which in local coordinates is wn = n! ( R) n det(go.,B)dz l

1\

di l

1\ ... 1\

... 1\

dz n 1\ din.

The volume is

(2.11)

Vol (M) = Volg (M) =

~ {

n·JM

The total scalar curvature may be written as (2.12)

w (n times) and

wn .

2.

CONNECTION, CURVATURE, AND COVARIANT DIFFERENTIATION

Let V i3 (2.13)

~

67

V 8 / 8z i3' The Laplacian acting on tensors is given by

1 ~ ~ "2l~/3 (VOtVi3

+ Vi3VOt)

1 ="2 (VOtVa

+ VaVOt).

NOTATION 2.17. At this point we have begun to use the extended Einstein summation convention where not only pairs of repeated indices consisting of an upper index and a lower index are summed but also pairs of repeated lower indices consisting of a barred index and an unbarred index are summed. For example, aOta ~ L:~,/3=1 gOti3aOti3' Formulas which are expressed using this convention hold in the literal sense in unitary holomorphic coordinates at a point p. Usually we use the extended Einstein summation convention in lieu of computing in unitary coordinates at a point. In this sense, the formulas in (2.7) and throughout this chapter hold in arbitrary holomorphic coordinates. EXERCISE 2.18. Show that the Laplacian defined above in (2.13) is onehalf of the Riemannian Laplacian. Acting on functions, the Laplacian is A

u =

g

Oti3n v

n 01 V

i3 = g

Oti3

since, acting on functions, VOl V i3 = 8z~~z~ is a real-valued function on M, then (2.14)

{j2

8z Ot 8z/3 '

= V i3V

01'

We also note that if ¢

~ Igrad¢1 2 = gOti3vOt¢Vi3¢ ~ IV¢1 2 ,

where the LHS is the Riemannian gradient. Likewise, (2.15)

1 2 = 1V OtVi3¢ 12 "2IHess¢1

+ IV

Ot

V/3¢1 2,

where the LHS is the Riemannian Hessian. In our calculation of evolution equations under the Kahler-Ricci flow we shall often use the following. LEMMA 2.19 (Kahler commutator formulas). On a Kahler manifold we have the following commutator formulas for covariant differentiation acting on tensors of type (1,0), (0,1), and (1,1) , respectively: (2.16) (2.17) (2.18)

V 01 V i3a-y - V i3 V Ota-y

= - R~i3-yao,

= -R~i3;ybiS = R~a}iS' VOl V i3 a-yiS - V i3 V Ota-yiS = - R Ot i3-yf/aT/iS + R Oti3T/iS a-yf/' VOl V i3 b;y - V i3V Otb;y

Analogous formulas hold for higher degree tensors and forms (see Exercise 2. 22).

REMARK 2.20. Note that ROti3T/iS = -ROt i3iST/' which exhibits the consistency of the formula above with the analogous formula in Riemannian geometry. We have also used the extended Einstein summation convention.

2. KAHLER-RICCI FLOW

68

Using the fact that the Christoffel symbols are zero unless all of the indices are unbarred or all are barred, we compute PROOF.

and

Va V /Ja'Y = OaO/Ja'Y - r~'Y0/Jao,

r~'Yao) .

V 13 Vaa'Y = 013 (Oaa'Y Hence, using (2.4), we have

Va V /Ja'Y - V13 V aa'Y = o/Jr~'Yao = - R~/J'Yao, which is (2.16). Equation (2.17) is the complex conjugate of (2.16). If a is a (1, I)-tensor, then we compute V /Ja'Y1, = 0/Ja'Y1, - r~1,a'Ye,

Vaa'Y1,

r~'YaTJ1,

= Oaa'Y1, -

and

Va V/Ja'Y1, = Oa (o/Ja'Y1, V 13 Vaa'Y1,

r~1,a'Ye)

-

r~'Y (o/JaTJ1, - r~1,aTJe)

,

= 013 (Oaa'Y1, - r~'YaTJ1,) - r~1, (Oaa'Ye - r~'YaTJe) .

Hence, after some cancellations, we obtain in holomorphic coordinates {za} at a point p where it is unitary,

Va V /Ja'Y1, - V13 Vaa'Y1, = -Oar~1,a'Ye + 0/Jr~'YaTJ1, = Ra/Je1,a'Ye - Ra/J'YfiaTJ1,'

o LEMMA 2.21 (Va and V (3 commute). We have [Va, V{3] = 0 acting on tensors of any type. That is, if a is a (p, q)-tensor, then

Va V{3a'Yl "''Y1'1,l ... 1,q - V{3 V aa'Yl "''Y1'1,l ...1,q = O. PROOF.

We have

V{3a 'Yl"''Y1'Ol'''Oq - - -- o{3a'Yl···'Y1'Ol···Oq - - -

r TJ a - {3'Yj 'Yl"''Yj-1TJ'Yj+l''''Y1'dl'''Oq '

It is easiest to compute the second covariant derivative in normal holomorphic coordinates centered at any given point p E M, where r~{3 (p) = O. In these coordinates we have at p,

Va V{3a'Yl ···'Y1'1,1···1,q = Oa (0{3a'Y1···'Y1'1,l ... 1,q -

- a o{3a -

a

- - -

'Y1"''Y1'd1'''dq

r~'Yj a'Yl "''Yj-1 TJ'Yj+1 "''Y1'1,1 ...1,q)

(a r a

TJ ) {3'Yj

a'Y1"''Yj-1 TJ'Yj+1"''Y1'01'''Oq' - -

2. CONNECTION, CURVATURE, AND COVARIANT DIFFERENTIATION

69

Hence V' a V'(3a,Yl' "'Yp81 .. ·8q

-

V' (3 V' aa'Yl .. ''Yp81 ···8q

-- (8(3r'1/a'Yj - {)a r'1/(3'Yj ) a'Yl···'Yj-l'1/'Yj+1···'Yp,h···8q· - -

Now from (2.3) we have at p,

8(3r~'Yj

-

8ar~'Yj

= g'1/5.. (8(38ag'Yj5.. - 8a8(3g'Yj5..) = 0,

o

and the lemma follows.

EXERCISE 2.22. Compute the commutator formula for the operator [V' a, V',a] ~ V' a V',a - V',aV' a acting on (p, q)-tensors and forms. SOLUTION TO EXERCISE 2.22. If a is a (p, q)-tensor, then V' a V' ,aa'Yl"''Yp81 ... 8q

-

V',a V' aa'Yl ''''Yp81 •.. 8q

p

L R a,a'Yifj a'Yl···'Yi-l'1/'YHl···'Yp8 ···8

=-

1

q

i=l

q

+L

Ra,a'1/8j a'Yl···'Yp81 ···8j _ 1 fj8j+l···8q·

j=l

The commutator [~, V',a] , acting on functions, is ~ V',af

1

= "2 (V' a V'ii + V'ii V' a) V',af =

1

1

"2 V' a V',a V'iif + "2 V',a V' ii V' af 1

= "2R'Y,aV'''ri + V',a~f.

(2.19)

LEMMA 2.23 (Kahler V' and ~ commutator). The commutator [V' a, ~l ,

acting on (0, I)-forms, is given by (2.20)

1

1

V'a~a,a - ~V'aa,a = "2V'a R8,aa8 - "2Ra8V'8 a,a + R a;Y8,aV''Y a8'

PROOF. We compute

+ V';y V' 'Y) a,a - (V''Y V';y + V';YV''Y) V' aa,a = V''Y (V' a V';y - V';y V' a) a,a + (V' a V';y - V';y V' a) V''Ya,a = V''Y (R a;Y8,aa8) - R a;Y'Y8V'8 a,a + R a;Y8,aV''Y a8 = V' aR8,aa8 - Ra8 V' 8a,a + 2R a;Y8,a V' 'Y a8'

2 (V' a~a,a - ~ V' aa,a) = V' a (V' 'Y V';y

where we used [V'a'V''Yl

= 0 and V''YRa;Y8,a = V'aR8,a from (2.9).

0

REMARK 2.24. By taking the complex conjugate of the lemma above,

we have (2.21)

2.

70

KAHLER-RICCI FLOW

Since the Kahler-Ricci flow is a heat-type equation for Kahler metrics, some evolution equations we shall derive later in this chapter use the following commutator formula. If a is a time-dependent (0, I)-form, then under the Kahler-Ricci flow,

[:t

-~, \/0:] a{J = ~\/ o:R8{Ja8 - ~RO:8\/8a{J + RO:18{J\/"(a8'

Another basic formula is the commutator of the heat operator and the Hessian. First, by (2.19) and (2.20), we have for any function f on M,

\/ 0: \/ {J~f

= ~L \/0: \/ {Jf ~ ~ \/ 0: \/{Jf

(2.22)

+ Ro:{J81 \/"( \/8f -

1 1 2 R o:8\/8\/{Jf - 2R"({J\/ 0: \/1f,

where ~L is called the (complex) Lichnerowicz Laplacian. If function also of time, then (2.22) tells us \/ 0: \/ {J (:t -

(2.23)

~)

f = (%t -

~L )

f is a

\/0:\/{Jf.

Note that since \/0: \/ {J = 8o:8f3 acting on functions, we have when acting on functions.

3t (\/0:\/(J) = 0

REMARK 2.25. Formula (2.23) has the following Riemannian analogue. For a time-dependent function f and under the Ricci flow, we have

(2.24) where

\/i\/j (%t ~L,

~)

f = (%t -

~L) \/i\/jf,

defined by ~LVij ~ ~Vij

+ 2Rkijf.Vkf. -

~kVjk

- RjkVik,

is the (Riemannian) Lichnerowicz Laplacian acting on symmetric 2-tensors (see Lemma 2.33 on p. 110 of [111]). In fact, (2.24) is a special case of (2.23). 3. Existence of Kahler-Einstein metrics In this section we discuss what is known about the existence and uniqueness of Kahler-Einstein metrics, which are canonical (constant Ricci curvature) metrics on Kahler manifolds. The Kahler-Einstein equation is elliptic whereas the Kahler-Ricci flow, discussed beginning in the next section, may be considered as its parabolic analogue. First recall the following 88-Lemma, which is a consequence of the Hodge decomposition theorem. LEMMA 2.26 (d-exact real (1, I)-form is 88 of real-valued function). Let M be a closed Kahler manifold. Ifw is an exact real (1, I)-form, then there exists a real-valued function 'IjJ such that Ff.88'IjJ = w. That is, n

82 8zo:8zf3 'IjJ

= wo:{J,

3.

EXISTENCE OF KAHLER-EINSTEIN METRICS

where W = HWo./3dzo. 1\ dz/3 and wo./3

71

= W/3ii'

PROOF. This is a standard result in the theory of Kahler manifolds; see the book by Zheng [383] for example. 0 REMARK 2.27. More generally, we have the following. Let b be a (p, q)form, where p, q > O. If b is d-closed and either d-, 8-, or 8-exact, then there exists a (p - 1, q - I)-form

(2.25)

P = 27rw.

Since Po H Rc (Yo) 0./3 dzo. 1\ dz/3 is a real (1, I)-form in the same cohomology class 27rCl (M) as 27rW = 27rHwo./3dzo.l\ dz/3, by Lemma 2.26 there exists a real-valued function f such that Rc (Yo)o./3 - 27rwo./3

= 80.8/3J.

Therefore equation (2.25), which in local coordinates is Ro./3 equivalent to

80.8/3f

= 27rwo./3' is

= Rc (yo)o./3 - Ro./3 2

2

8 ) 8 = - 8 z 0.8z/3 log det ( (Yo)'Y8 + 8 z 0.8z/3 log det (Y'Y8) . Since M is closed, this implies det

(y 8)

(1')

log

det (Yo)'Y 8

=

f + log C

for some constant C > O. Since the real (1, I)-forms wo and ware in the same cohomology class, using Lemma 2.26 again, we see that there exists a real-valued function

Y'Y8

= (Yo)'Y 8 + 8'Y 86
Thus we may rewrite (2.25) as a complex Monge-Ampere equation: det ((yo)'Y 8 + 8'Y86

----'---:---~-'- - C ef

det ((yo)'Y 8)

-

.

72

2.

KAHLER-RICCI FLOW

Yau's proof of Theorem 2.28 involves solving the fully nonlinear equation above by using the continuity method. The proof was a tour de force. One says that a Kahler metric 9 is Kahler-Einstein if p = >.w for some >. E R If M n admits a Kahler-Einstein metric g, then

(We have>. = ~, where the (complex) scalar curvature R is constant.) Therefore, a necessary condition for the existence of a Kahler-Einstein metric on M is that its first Chern class have a sign. By having a sign we mean that ct (M) = 0, < 0, or > 0 if there exists a real (1, I)-form in the first Chern class which is zero, negative definite, or positive definite, respectively. COROLLARY 2.29 (Cl = 0: existence of Kahler Ricci flat metrics). If (Mn, go) is a closed Kahler manifold with Cl (M) = 0, then there exists a Kahler metric 9 with [w] = [wo] such that Rc (g) == O. Kahler Ricci flat metrics are called Calabi-Yau metrics and Kahler manifolds with Cl (M) = 0 are called Calabi-Yau manifolds. Another consequence of Theorem 2.28 is that if Cl (M) < 0 (respectively, Cl (M) > 0), then in each Kahler class there exists a metric with negative (respectively, positive) Ricci curvature. When Cl (M) < 0, we have the following result about the existence of Kahler-Einstein metrics conjectured by Calabi and proved by Aubin [11,12] and Yau [378, 379]. Calabi proved that such a Kahler-Einstein metric is unique if it exists [43]. THEOREM 2.30 (Cl < 0 Calabi conjecture: R < 0 Kahler-Einstein metrics). If Mn is a closed complex manifold with ct (M) < 0, then there exists a Kahler-Einstein metric 9 on M, which is unique up to homothety (scaling), with negative scalar curvature. A consequence of Theorem 2.30 is the following Chern number inequality:

~

(-It 2 (n + 1) cl(M)n-2 c2 (M). n (See Yau [378] and also Corollary 9.6 on p. 226 of [383] for an exposition.) (-It cl(Mt

REMARK 2.31. Another consequence of Theorem 2.30 is that if (M2, g) is a closed Kahler surface homotopically equivalent to CP2, then M2 is biholomorphic to Cp2 (see Yau [378]; earlier related work in all dimensions was done by Hirzebruch and Kodaira [204]). If Cl (M) > 0, however, there are obstructions to the existence of KahlerEinstein metrics. An example is the Futaki invariant (see [147]). On a closed manifold Mn with Cl > 0 (Le., a Fano manifold), fix a Kahler metric 9 such that [w] is a positive real multiple of cl(M). (This is the so-called

3. EXISTENCE OF KAHLER-EINSTEIN METRICS

73

canonical case.) By scaling the metric, we may assume [w] = cl(M). By Lemma 2.26 there exists a smooth function f : M --+ IR such that p-

27l'W

=

Raar

(One can make f unique by the normalization fM e- f d/-t = 1.) Let 1}(M) denote the space of (real) holomorphic vector fields on M. The Futaki functional F[wJ : 1}(M) --+ C is defined by

F[wJ (V)

~ 1M V (I) d/-t = 1M (V, \7 f)

d/-t.

Futaki [147] showed that F[wJ is well defined, i.e., that it depends only on the homology class [w]. It is then clear that if M admits a Kahler-Einstein metric, then F[wJ vanishes. However, Tian has shown that 1}(M) = 0 (which implies F[wJ = 0) does not imply there exists a Kahler-Einstein metric. The following uniqueness result in the Cl > 0 case was proved by Bando and Mabuchi [21]. THEOREM 2.32 (Uniqueness of Kahler-Einstein metrics). Let (Mn,g) be a closed Kahler manifold with Cl (M) > O. The Kahler-Einstein metric (with positive scalar curvature), if it exists, is unique up to scaling and the pull back by a biholomorphism of M.

The following result was proved by Andreotti and Frankel [144] for n = 2, Mabuchi [260] for n = 3, and Mori [271] and Siu and Yau [336] in all dimensions; Mori proved a more general algebraic-geometric result. Work on characterizing cpn was done by Kobayashi and Ochiai [237]. THEOREM 2.33 (Frankel Conjecture). If(Mn, g) is a closed Kahler manifold with positive bisectional curvature, then M n is biholomorphic to cpn.

In fact, if the bisectional curvature is nonnegative everywhere and positive at some point, then M is biholomorphic to cpn. When n = 2 and Cl (M) > 0, we have the following (see Tian [345]). 2.34 (Cl > 0 surfaces: R > 0 Kahler-Einstein metrics). If M2 is a closed complex surface with Cl (M) > 0 and the Lie algebra of the automorphism group is reductive, then there exists a Kahler-Einstein metric g on M with positive scalar curvature. THEOREM

2.35. Note that such surfaces are biholomorphic to CP2 blown up at p points, where 3 ~ p ~ 8. On the other hand, for n ~ 2, cpn blown up at 1 or 2 points does not admit a Kahler-Einstein metric (see p. 156 of Lichnerowicz [254] and Yau [376]). See Section 7 of this chapter for the existence of Kahler-Ricci solitons on CP2 blown up at 1 or 2 points. REMARK

There are a number of additional works related to stability and the existence of Kahler-Einstein metrics with Cl > O. Notably Aubin [14], Siu [335]' Nadel [281]' Tian [346], Donaldson [129], [130]' and Phong and

74

2. KAHLER-RICCI FLOW

Sturm [304]. The existence of Kahler-Einstein metrics on complete noncompact manifolds with Cl < 0 and Cl = 0 has been well studied (see Cheng and Yau [94] and Tian and Yau [349], [350] for example). For work on the existence of singular Kahler-Einstein metrics on certain classes of closed Kahler manifolds where Cl does not have a sign, see Tsuji [360] (for some further recent work see Cascini and La Nave [60] and Song and Tian [337]). Kahler-Einstein metrics are closely related to Kahler-Ricci solitons and hence the Kahler-Ricci flow which will be discussed next in this chapter.

4. Introduction to the Kahler-Ricci flow In this section we introduce the Kahler-Ricci flow system and its equivalent formulation as a single parabolic Monge-Ampere equation. We discuss some basic estimates which may be proved using the maximum principle. 4.1. The Kahler-Ricci flow equation. Let (Mn, J) be a closed manifold with a fixed almost complex structure. Given a Riemannian metric g, we may define a 2-tensor w by w (X, Y) ~ 9 (JX, Y). Recall that when 9 is Hermitian, w is antisymmetric (Le., defines a 2-form) and w is called the Kahler form. If a solution 9 (t) to the Ricci flow ~gij = -2~j is Hermitian at some time t, then w satisfies the equation gtW = -2p at that time, where p = p (t) is the Ricci form of 9 (t) . Hence

~ (dw) = d (~w) at at

= -2dp = 0

whenever 9 (t) is Kahler (we can define even when 9 is not Hermitian). This suggests that if 9 (0) is Kahler, then under the Ricci flow 9 (t) is Kahler for all t 2: O. Consider the Kahler-Ricci flow equation (2.26)

a

at gai3

= -Rai3

for a 1-parameter family of Kahler metrics with respect to J, which is obtained from the Ricci flow by dropping the factor of 2. Now we derive the parabolic complex Monge-Ampere equation to which the Kahler-Ricci flow is equivalent. For a complete initial Kahler metric with bounded curvature, we will use this scalar equation to prove the short-time existence of a solution to the initial-value problem for the Kahler-Ricci flow. On a closed manifold we will use the scalar equation to prove that the Kahler property of an initial metric is preserved under the Ricci flow and to prove the long-time existence of solutions to the Kahler-Ricci flow. By (2.26), gt [w] = - [p (t)] = - [p (0)] , so that the Kahler class of the metric at time t evolves linearly,

[w (t)] = [w (0)] - t [p (0)] , and the real (1, 1)-forms w (t) - w (0) + tp (0) are exact for t 2: O. Let g~i3 ~ gai3 (0). Using Lemma 2.26, for each t there exists a real-valued function

4.

INTRODUCTION TO THE KAHLER-RICCI FLOW

75

'-P (t) defined on all of M such that o

-

0

-

ga:fi (t) = ga:!3 + tOa:o,a log det g')'6 + oa:o,a'-P (t) .

(2.27)

By (2.6) we have

Hence, by differentiating (2.27), we obtain

Oa: 8,a (:t '-P) = - Ra:fi - oa: 8,a log det g~6 _

det

(g~6 + to')'8olog det g~iI + o')'8o'-P (t) )

= oa:o,a log

dO.

etg')'6

Hence we conclude that the Kahler-Ricci flow equation on a closed manifold is equivalent to the following parabolic (scalar) complex MongeAmpere equation: (2.28)

o'-P

~ = log

vt

det

(g~6 + to-y8o log det g~iI + o')'8o'-P (t») d

0

etg')'6

+ Cl (t)

for some function of time Cl (t) . By standard parabolic theory, given any Coo initial function '-Po on a complete Kahler manifold with bounded bisectional curvature, there exists a unique solution '-P (t) to (2.28) with '-P (O) = '-Po, defined on some positive time interval 0 ~ t ~ E. We also have the following. LEMMA 2.36 (The Kahler property is preserved under the Ricci flow). If (M n , J, go) is a closed Kahler manifold, then there exists a solution to the Kahler-Ricci flow 9 (t), 0 ~ t ~ E, for some E > 0 with 9 (0) = go. Furthermore 9 (2t) is a solution of the (Riemannian) Ricci flow. Also any solution 9 (t) of the (Riemannian) Ricci flow with 9 (t) = go must be Kahler (preserving the compatibility with the almost complex structure). PROOF. Given go, we can find a solution '-P (t), 0 ~ t ~ E, of (2.28) with (t) == O. From the derivation of (2.28), we know that 9 (t) defined by (2.27) is a solution of (2.26). Hence 9 (2t) is a solution of the Ricci flow. The last statement follows from the uniqueness of the initial-value problem for the Ricci flow. D Cl

REMARK 2.37. From the derivation of (2.28) it is clear that if we have a bounded C 4-solution '-P (t) for some Cl (t) on any complex manifold (regardless of completeness and compactness), then we get a C 2 -solution 9 (t) defined by (2.27) to the Kahler-Ricci flow.

2. KAHLER-RICCI FLOW

76

4.2. The normalized Kahler-Ricci flow equation. Let (Mn, J, go) be a closed Kahler manifold. Now we make the basic assumption (corresponding to the canonical case), holding for the rest of this section and the next section, that the first Chern class is a real multiple of the Kahler class, i.e., that

[pol

=

c[wol

for some c E JR. Note that this is possible only if the first Chern class has a sign, i.e., is negative definite, zero, or positive definite. Comparing (2.11) and (2.12), we find that c = ~, where r ~ 1M Rodj.Lga/ Volga (M) is the average (complex) scalar curvature, so that

r 1 -2 [wol = -2 [pol = 7rn

7r

Cl

(M) .

So r depends only on the cohomology class [wol, n, and The normalized Kahler-Ricci flow is

o

otga~ = -Ra~

(2.29)

r

+ ;:;,ga~'

Cl

(M).

for t E [0, T).

The solution of (2.29) can be converted to the solution (2.26) by scaling the metric and reparametrizing time, and vice versa (see Section 9.1 in Chapter 6 of Volume One or subsection 9.1 below). Hence, from Lemma 2.36, we know that the initial-value problem for (2.29) with 9 (0) = go has a solution for a short time. By a derivation similar to that of (2.28), we get the following parabolic (scalar) complex Monge-Ampere equation, corresponding to (2.29) with ga~ (t) = g~~ + Oa8f3'P (t) ,

(2.30)

o'P ot

det

(g~J + O-y 80'P )

= log

detg O-

r +;:;,'P - 10 + Cl (t)

'"(0

for some function of time

Cl

(t). Here 10 is defined by Ra~ (go) - ~g~~ =

Oa8f310; this is possible because [-po

+ ~wo] = O.

4.3. Basic evolution equations. Let 9 (t) be a solution of either the Kahler-Ricci flow or the normalized Kahler-Ricci flow. We define the potential function 1 = 1 (t) by (2.31) This equation is solvable since [- p + ~w] = 0 and by Lemma 2.26. Note that 1 is determined up to an additive constant. Taking the trace of (2.31), we have (2.32)

R-r='~.f.

4.

INTRODUCTION TO THE KAHLER-RICCI FLOW

77

Differentiating (2.3), we find that for both the Kahler-Ricci flow and the normalized Kahler-Ricci flow, the Christoffel symbols evolve by

8 r"{ at Ot{3 -

(2 .33)

"{8'r7 R -

-9

VOl.

{30'

The volume form and scalar curvature evolve according to the following. LEMMA 2.38 (Evolution of df-L and R for normalized flow). Under the normalized Kahler-Ricci flow (2.29),

8 8t df-L = (r - R) df-L and

-8R = flR + 1R 01.{3- 12 - -nr R. 8t

(2.34)

In particular, since fM (r - R) df-L = 0, the normalized Kahler-Ricci flow preserves the volume. PROOF. We first compute, using (2.29), that

8

(2.35)

88

8t log det 9"{8 = 9 "{ 8t 9"{8 = r - R.

Hence

8 8t df-L = (r - R) df-L.

The evolution of the Ricci tensor is (2.36)

8

- (88t

8t ROt/3 = -8Ot8{3

log det 9"{8

)8Ot8{3R. =

From this and

8R _ 01./3 8

at -

9

_

8

_

_

at ROt{3 - at 9Ot{3' ROt{3

D

we easily derive (2.34).

EXERCISE 2.39 (Evolution of R for unnormalized flow). Show that under the Kahler-Ricci flow %t 9Ot/3 = -ROt/3' we have %tdf-L = -Rdf-L, %tROt/3 = 8Ot8{3R, and

(2.37)

8 R -_ at

9

01./3 at 8 ROt{3_ + 1R {3_12 -_ flR Ot

+ 1R Ot{3_1 2 .

EXERCISE 2.40 (Total and average scalar curvature evolution). Show that if M n is closed, then under the Kahler-Ricci flow ft9Ot/3 = -ROt/3'

! 1M

Rdf-L =

1M (I ROt/312 - R2) df-L,

and hence we have

~: = (1M df-L) 1M (I ROt/312 -1

R2) df-L + r2.

78

2.

KAHLER-RICCI FLOW

REMARK 2.41 (I-dimensional normalized Kahler-Ricci flow). Recall the following facts, due to Hamilton [180], about the normalized Ricci flow on Riemannian surfaces £g = (r - R) g. Throughout this remark, Rand r denote the Riemannian scalar curvature and its average, respectively. Note that 9 (t) ~ 9 (!t) is a solution of the complex I-dimensional normalized Kahler-Ricci flow. The potential function f, defined by (2.32) and normalized suitably by an additive constant, satisfies (see Lemma 5.12 on p. 113 of Volume One) of (2.38) ot = b.f + r f. By the maximum principle, this implies If I ~ Ce Tt • The gradient quantity H ~ R - r + 1\7 fl2 satisfies (see Proposition 5.16 on p. 114 of Volume One) oH (2.39) at = b.H - 21MI2 + rH, where M ~ \7\7 f - !b.f· g. By the maximum principle, we have (see Corollary 5.17 on p. 115 of Volume One)

_CeTt ~ R - r ~ H ~ CeTt . This gives the exponential decay of IR - rl when r < O. The norm squared of the tensor M evolves by (see Corollary 5.35 on p. 130 of Volume One)

:t

(2.40)

IMI2 = b.IMI2 - 21\7 MI2 - 2R IMI2 .

Generalizing the I-dimensional formula (2.38) to higher dimensions, the potential satisfies a linear-type equation. (Strictly speaking, the equation is not linear since the Laplacian is with respect to the evolving metric.) LEMMA 2.42 (The potential f satisfies a linear-type equation). Under the normalized Kahler-Ricci flow on a closed manifold Mn, the potential function f, defined by (2.31) and normalized by an additive constant, satisfies (2.41 )

PROOF. From (2.31) we compute On[)(3

(it) =

:t (on[)(3f)

= on o-(3R -

=

! (Rn~

r0

-

~gn~)

- (

r) .

-:;;, otgn~ = on o (3 b.f + -:; , f

Since M is closed, it follows that (2.42)

~~ =

b.f + ~f + c(t)

for some function c (t) and the lemma follows from the fact that we have the freedom of adding a time-dependent constant in our choice of f (x, t). 0

4.

INTRODUCTION TO THE KAHLER-RICCI FLOW

79

COROLLARY 2.43 (Estimate for f). If Mn is closed, then, for the function f given by Lemma 2.42, we have If I :S Ge;;t.

(2.43)

This is the first hint that the Kahler-Ricci flow in the case where Cl (M) < o is the easiest and that the case where Cl (M) > 0 is the hardest. We now compute for f in Lemma 2.42, for the normalized Kahler-Ricci flow, that (2.44)

%t

IV' c.d1 2= illV'afl 2 - IV'a V' ,8f1 2 - IV'a V' ,6f1 2 + ~ IV'afl 2 .

Define

h ~ ilf + IV' afl 2 = R - r Similarly to (2.39), we have

+ IV'afl 2 .

LEMMA 2.44 (Ricci soliton gradient quantity evolution). For the normalized Kahler-Ricci flow on a closed manifold Mn,

~~

(2.45)

= ilh - IV'a V' ,8f1 2 + ~h.

PROOF. We compute

{) (R at (2.46)

r) =

il (R -

r)

+ IRa,6 12 -

=

il (R -

r)

+ lV' a V',6fl + -ilf + -n n

=

il (R -

r)

+ IV'a V' ,6f1 2 + :cn (R -

rR -:;;, 2

2r

r2

r --R n

r),

where we used (2.31) and (2.32). Equation (2.45) follows from summing this equation with (2.44). 0 COROLLARY 2.45 (Estimate for R). rt

rt

-Ge n :SR-r:SGe n ,

(2.47)

lV'fI2:s Ge;;t.

(2.48)

PROOF. By (2.46), the lower bound for R - r follows from

(!

-il) (R-r)

~ ~(R-r).

To get the upper bound for R-r, we observe that by the maximum principle, we have rt R - r :S h:S Gen. This also implies (2.48) since IV' fl2 = h - (R - r) :S h + Ge;;t. 0 REMARK 2.46 (Exponential decay when Cl < 0). When Cl (M) < 0, so that r < 0, (2.47) says that R approaches its average exponentially fast. This suggests that the Kahler-Ricci flow converges to a Kahler-Einstein metric. Indeed, this is Theorem 2.50 below.

2. KAHLER-RICCI FLOW

80

Here is an interesting equation due to Hamilton. LEMMA 2.47 (Ricci soliton vanisher evolution equation). For the potential function f in Lemma 2.42, we have

(2.49)

%t IV' 0 V' /3f12 =

~ IV' 0 V' /3f12 -

IV' "I V' 0 V' /3f12 - lV'i V' 0 V' /3f12

- 2 RoihJ V' a V'i fV' /3 V' 8f. PROOF.

By (2.42) and the commutator formulas, we have

%t (V'oV'rd)

=

V' 0 V'/3

= V'0V'/3

(~{) - (%tr~/3) V''Yf (~f + ~f) + V'oRtJiV''Yf.

On the other hand, for any function V'oV'/3~f =

f,

V'oV'/3V''YV'if = V''YV'oV' i V'/3f

= V' /v i V' 0 V' /3f - V'"I (Roi!3JV' 8f) =

1

'2 (V' "I V'i + V'i V' "I) V' 0 V' /3f -

V' oR/3JV' 8f - ROi/3JV' "I V'/d

1

- '2 (ROi V'"I V' /3f + RtJi V' 0 V'"If) since Hence

a

at (V' 0 V' /3f) = ~ V' 0 V' /3f

r

+ -;;, V' 0 V' /3f -

Roi!3JV' "I V' 8f

1

- '2 (ROi V' "I V' /3f + R/3i V' 0 V'"If)

o

and one easily derives (2.49) from this.

2.48. Find geometric applications of (2.49) in the study of the Kahler-Ricci flow. PROBLEM

EXERCISE

2.49. Show that when dime M = 1,

1V'0V'/3fI2 =

~ lV'iV'jf - ~~f9ijI2

and

Show also that (2.49) generalizes (2.40). 5. Existence and convergence of the Kahler-Ricci flow In this section we present some of the proofs of the basic global existence and convergence results for the Kahler-Ricci flow due to H.-D. Cao [46].

5.

EXISTENCE AND CONVERGENCE

81

5.1. Cao's existence and convergence theorem. When the first Chern class has a definite sign, either negative, zero, or positive, Cao proved that the normalized Kahler-Ricci flow exists for all time. Let KRF and NKRF denote the Kahler-Ricci flow and the normalized Kahler-Ricci flow, respectively.

THEOREM 2.50 (NKRF: C1 <, =, > 0 global existence). Let (Mn,go) be a closed Kahler manifold with

(1) either C1 (M) < 0, C1 (M) = 0, or C1 (M) > 0, and (2) r;;- [wo] = 21l" C1 (M). Then there exists a unique solution g(t) of the normalized Kahler-Ricci flow defined for all t E [0,00) with g(O) = go. When the first Chern class is nonpositive, Cao proved that the normalized Kahler-Ricci flow converges to a Kahler-Einstein metric in the same

Kahler class as the initial metric. THEOREM 2.51 (KRF: C1 SO convergence). Let 9 (t) be a solution of the normalized Kahler-Ricci flow, as in Theorem 2.50, with ct (M) S o. Then 9 (t) converges exponentially fast in every Ck-norm to the unique KahlerEinstein metric goo in the Kahler class [wo]. Note that the initial-value problem for the normalized Kahler-Ricci flow equation

o

r

otga{J = -Ra{J + ;,ga{J' ga{J (0) = g~{J' is equivalent to the following parabolic Monge-Ampere equation for the metric potential function
o

r

+ -
f(x, 0),

where (2.52)

and f(x,O) is the potential function of Ra{J(x, 0) - ~ga{J(x, 0) defined by (2.31). To prove Theorem 2.50 and Theorem 2.51, it suffices to prove the long-time existence of
5. EXISTENCE AND CONVERGENCE

81

5.1. Cao's existence and convergence theorem. When the first Chern class has a definite sign, either negative, zero, or positive, Cao proved that the normalized Kahler-Ricci flow exists for all time. Let KRF and NKRF denote the Kahler-Ricci flow and the normalized Kahler-Ricci flow, respectively. THEOREM 2.50 (NKRF:c1 <,=,> 0 global existence). Let (Mn,go) be a closed K iihler manifold with

(1) either C1 (M) < 0, C1 (M) = 0, or C1 (M) > 0, and (2) ~ [wol = 27TC1 (M). Then there exists a unique solution g(t) of the normalized Kahler-Ricci flow defined for all t E [0,00) with g(O) = go. When the first Chern class is nonpositive, Cao proved that the normalized Kahler-Ricci flow converges to a Kahler-Einstein metric in the same Kahler class as the initial metric. THEOREM 2.51 (KRF: C1 ::; 0 convergence). Let 9 (t) be a solution of the normalized Kahler-Ricci flow, as in Theorem 2.50, with C1 (M) ::; O. Then 9 (t) converges exponentially fast in every Ck-norm to the unique KahlerEinstein metric goo in the Kahler class [wol.

Note that the initial-value problem for the normalized Kahler-Ricci flow equation

8

r

atgai3 = -Rai3 + -:;;,gai3' gai3 (0)

= g~i3'

is equivalent to the following parabolic Monge-Ampere equation for the metric potential function
8

r

+ -
f(x, 0),

where (2.52)

and f(x,O) is the potential function of Rai3(x, 0) - ~gai3(x, 0) defined by (2.31). To prove Theorem 2.50 and Theorem 2.51, it suffices to prove the long-time existence of
2. KAHLER-RICCI FLOW

82

Moser iteration, and differential Harnack estimates (see the discussion below). The corresponding theory for systems is considerably harder, sometimes tractable only under more restrictive conditions such as the nonnegativity of the bisectional curvature (see Sections 6-8 of this chapter). Since (2.53)

we have that

v(x, t)

.

7 -

8cp

at (x, t)

is also a potential function of Ro:fj(x, t) - ~9o:fj(X, t). From (2.50), (2.53), and (2.51), we compute

8v _ 8 (8CP) _ o:fj 8 _ r 8cp 8t - - 8t 8t - -9 8t 9o:fj - -:;;, 8t r n

= 6.v +-v

with the initial condition v(x,O) = f(x,O). Therefore, if we insist, as in Lemma 2.42, that the potential function f(x, t) satisfies the heat equation -& f = 6.f + !:. f, we must have n

f = _ 8cp 8t .

(2.54)

Recall from (2.43) that following.

ifi

I~ I ::;

Ce:;'t. More precisely, we have the

LEMMA 2.52 (Time-derivative estimate for cp). r t 8cp r t -Clen ::; f (x, t) = - 8t ::; C2 en ,

(2.55)

where C I

~ -

minxEMn f(x, 0) and C 2 ~ maxxEM f(x, 0).

5.2. Proof of Theorem 2.50. Theorem 2.50 is proved via a progression of estimates which culminates with a C2,O:-estimate for cp (t) on bounded time intervals. The CO-estimate is the following. LEMMA 2.53 (CO-estimate: bound for cp-uniform when r =I 0, then (2.56)

- C;n (e:;.t

-1) ::; cp(x,t)::; C:n (e:;.t -1).

Ifr = 0, then -C2t ::; cp(x, t) ::; CIt. PROOF. For the upper bound we compute

t

8cp (t r cp(x, t) = cp(x, 0) + Jo 8t (r) dr::; Jo Cle nT dr.

CI ::;

0). If

5. EXISTENCE AND CONVERGENCE

C:.n (e;-t -

If r =1= 0, the integral on the RHS is equal to obtain CIt. Similarly for the lower bound.

83

1) . When r

= 0, we 0

REMARK 2.54. The qualitative dependence of the estimate (2.56) on the sign of r should be compared to Corollaries 2.43 and 2.45.

The next estimate, a bound for the determinant of the complex Hessian of cp, is also a straightforward application of the previous result and the estimates (2.55) and (2.56) to equation (2.50). 2.55 (Estimates for the volume form-uniform when r ~ 0). If 0, then there exists a constant C ~ 1 such that

LEMMA

r

~

(2.57)

for all x E

~ < det (90 0, then _ en (eii-t-l) det (90
for all x E M and t PROOF.

~

O.

Applying (2.47) to (2.35), we have

~ log det (90
(2.58)

at

Hence, if r

=1=

Ir _ RI ~ Ce;-t.

=

det (90
0, then

log

det (90
~

Cn (.!.t ) en -1 , r

-

so that

In particular, if r < 0, then e

en r

t)) < det (9~a(X, ~/J

~

en

e - -;:- .

- det (90
When r

=

det g", -(x,t) det g",i3(x,O)

0, equation (2.58) is not strong enough to uniformly estimate . In this case we use

a

det (90
at

det (90
- log

=r -

R

af = -!:l.f = - -

at '

which implies log

det (90
f (x, t) + f (x, 0) .

2. KAHLER-RICCI FLOW

84

By the uniform bound (2.43) on

I,

we conclude

1 det (9ai3(X, t))
-::;-< for some

C~

o

1.

2.56. Alternatively, when r and (2.56) to equation (2.50), we have REMARK

log

det 9ai3(X, t) = det 9ai3(X, 0)

=1=

0, applying the estimates (2.55)

locp r I !:It (x, t) - -ncp(x, t) + I(x, 0) u

~ 11/(·,0)1100 (e;;t + le;;t - 11 + 1) ,

11/(·,0)1100' In particular, ifr < 0, then e- 2I1f (-,O)lIoo < det (9ai3(X, t)) < e2I1f (.,O)lIoo

since max{C1 ,C2 } ~

- det (9ai3(X, 0)) Lemma 2.55 is the first step towards proving that V a V i3CP is bounded and that 9ai3(X, t) is equivalent to 9ai3(x, 0), which in particular implies that 9ai3(X, t) is always positive definite. Next we estimate the trace of 9ai3(X, t) with respect to 9ai3(X, 0). Let

(2.59)

Y(x, t) ~ 9ai3 (X, 0)9ai3(X, t)

be the trace-type quantity we want to estimate. As we shall see below, a bound for Y (t) will imply a C 2-estimate for cp (t) . From (2.52) we have Y

(2.60)

= n + ~g(O)CP,

n = 9ai3 (t)9ai3(0)

+ ~g(t)CP.

Hence an estimate for Y implies an estimate for ~g(O)CP. Let Aa denote the eigenvalues of 9ai3(t) with respect to 9ai3(0). Then (2.61)

Y

= E~=lAa

and the eigenvalues of ( 8za:'tz13 ) with respect to 9ai3(0) are Aa -1. If Y ~ C, then as long as 9ai3(t) is positive-definite, Aa ~ C for each 0::. On the other hand, by Lemma 2.55, we have 1 n C ~ Aa ~ C, a=l where for r ~ 0, the constant C is independent of time, whereas for r > 0, C depends on time but remains bounded as long as the solution exists (though the bound may tend to 00 as time approaches 00). Hence there exists a constant c > 0 such that Aa ~ c, where c is independent of time for r ~ 0 and may depend on time for r > O. So indeed, 9ai3(t) remains positivedefinite as long as the solution exists and we have c ~ Aa ~ C', for some

II

5.

EXISTENCE AND CONVERGENCE

85

constant C' < 00. Thus a bound on Y shall imply an estimate of the complex Hessian of
for some C' <

00.

By abuse of notation, we shall call this the C 2 -estimate.

REMARK 2.57. A quantity similar to Y also played an important role in the later work of Donaldson [128] on the Hermitian-Einstein flow. We now turn to estimating Y. First we apply the heat operator to log Y. LEMMA 2.58. We have (2.63)

where

{)) 9"Y8(t)90
at

-

Y

r

+ -n'

~ ~ ~g(t) and R~/ (0) ~ 9/3"1 (0) R~8"1(0).

PROOF. From (2.59) and (2.29), we compute 1 {) -90
(2.64)

~I

+ !:.Y n

(2.65)

Given any point x E M, we will calculate in a local holomorphic coordinate system which is normal with respect to the metric 9(0) at x, so that i}a9/31 (x, 0) = 0 and 90
{)2

{)2

R~8J,1 (0) = _90
~Y =

9"Y8

{)2

{)z"Y{)z8

(90
_ "Y8 _ {)2 _() - 9 90
+9

"Y8 0
____

=

(2.66)

9"Y890
= 9"Y8R~/ (0)90
+9

0.fj {) {)_ 0 9 9 {)z8 90
where we used (2.5) and (2.67)

{)2

2. KAHLER-RICCI FLOW

86

(using the Kahler identities (2.2) to get the second equality in (2.67)). Note that

We claim that the Cauchy-Schwarz inequality gives (2.68)

IV'YI2 ~ Y

(g (ot.8 9"(8g0ji. ogo.8 OgVp ) oz"( oz8

.

By applying (2.68) to (2.66), we then obtain (2.65), as desired. To prove (2.68), we first observe that we may further assume go.8(x, t)

= >"0 80/3

is diagonal, where >"0 is defined above. The xas

=Y

LHS

of (2.68) can be written at

( (O)v(3 "(8 oji. ogo(3 OgVp) 9 9 9 oz"( oz8

.

Thus the claimed inequality (2.68) and the lemma are both proved.

0

Next we prove the key C 2-estimate (2.62) via an application of the maximum principle to (2.63). 2.59 (C2-estimate for
87

5. EXISTENCE AND CONVERGENCE

(1) (Uniform estimate when Cl ~ 0) If Cl (M) ~ 0, then there exists a constant C < 00 depending only on the initial metric such that

(2.69)

Y(x, t)

~

C

for all (x, t) EM x [0, T). Hence there exists C f < of T, such that the complex Hessian of cp satisfies

00,

independent

IVov~cp(t)lg(o) ~ C f

(2.70)

on M x [O,T). (2) (Time-dependent estimate when Cl > 0) If Cl (M) > 0, then there exist constants C and C f , both depending on 9 (0) and T < 00, such that the above estimates (2.69) and (2.70) hold on M x [0, T).

2.60. Estimate (2.70) implies IAg(O)cpl ~ C f • On the other hand, by Lemma 2.53, we have Icpl ~ Co for some Co < 00. Hence, by standard elliptic theory, we have for any a E (0,1), IIcpllcl.<> ~ Cl for some C 1 < 00 depending on a. However at this stage of the proof, it is not clear whether IHess cpl is bounded, where Hess denotes the real Hessian, but we do not need this. REMARK

By the discussion above, regarding inequality (2.62), we only need to bound Y from above. Again, we calculate in local holomorphic coordinates around x where go~(x, 0) = 80 {3, 8~"Tgo~(x, 0) = 0, and go~(x, t) = >'080{3. By (2.63) and >'0 ~ Y, we have PROOF.

(2.71)

( -a - uA) 1og Y < at

Roii"/'Y (0)

-

Y

>'0 + -r < C 1 ~ L.." -1 + -r >."/ n ,,/=1 >."/ n'

where Cl is a constant depending only on a lower bound of the bisectional curvatures of g(O). To control the bad terms on the RHS above, we consider equation (2.54) . (2.72)

(ata) - cp = - f A

Acp = -

1 L ~' n

f - n+

,,/=1

"/

where the second equality follows from (2.60). Consider the modified quantity w = logY - (C1 + l)cp. Combining (2.71) and (2.72), we have (2.73)

(ata) w<-""-+C2 - L.." >. ' --A

n

,,/=1

1

"/

where C 2 depends on C 1 and Ilflloo (if r ~ 0, then C 2 is independent of time, and if r > 0, then C 2 depends on time). By the maximum principle, (2.73) implies that w can be bounded above on M x [0, T) by a constant C depending on T.

88

2.

KAHLER-RICCI FLOW

To get a bound for w independent of T when r ~ 0, we need to work harder. When r ~ 0, we shall use the term - E"Y >..\ to dominate (from below) a function of w which approaches -00 as w ~ 00. Using equation (2.50), f = - %tc.p, and (2.61), we have

f (x, t) - f(x, 0)

det 9cr[J(X, t) (0) , et 9cr[J x,

r

+ -c.p(x, t) = -log d n

and hence

y e!(x,t)- !(x,O)+~
II : 2: (II : )~ (2: :a) 2: Acr . cr

=

a

"Y

"Y~a

"Y

n-l

a

"Y

using a standard inequality (we dropped a factor of nnl_2 since it makes the inequality easier to see). Since f and c.p are uniformly bounded, this implies that there exists a constant C3, which only depends on the initial data, so that

y

(2.74) Notice that

eW

=

~C ~

e-(Cl +l)
(2.75)

eW

3 (

:0)

n-J

We then have

~ (C4~

:J-J,

where C4 > 0 only depends on the initial data. Combining (2.73) and (2.75), we have

(ata) -

(2.76)

-

~

- 1 +C2 . w < - -1e n~ C4

-

This implies that, at any time t where W max (t) ~ (n - 1) log (C2C4 ) , we have 1tWmax (t) ~ O. Hence, by the maximum principle, sup w (x, t) xEfiA

~ max {sup w (x, 0), (n -

1) log (C2 C4 )}

xEfiA

for all t E [0, T), and hence Y is also bounded from above. When r ~ 0, both the constant C2 and the function c.p are uniformly bounded independent of T; hence Y is uniformly bounded independent of T. 0 Now we can complete the proof of Theorem 2.50. PROOF OF THEOREM 2.50. Let c.p (t), t E [0, T), be a solution of the NKRF (2.50) on a closed Kahler manifold (Mn, 90), where T is the maximal time of existence. If T < 00, then by the C 2 -estimate (Le., the estimate for the complex Hessian of c.p), the metrics 9(t) are uniformly equivalent to 9(0)

5. EXISTENCE AND CONVERGENCE

89

on the time interval [0, T). Once we have a uniform C2,a-estimate of cp on [0, T) (we shall prove the C 2 ,a- estimate in the next subsection), by choosing a time to < T close enough to T, the NKRF (2.50) with initial condition cp (to) can be solved on [to, to + E] , where Edepends on the C 2 ,a- estimate of cp but not to. If we choose to = T -~, this implies that we have a solution cp (t) for t E [0, T + ~]. This contradicts T being the maximal time of existence; hence the theorem is proved. D We shall supply the details for the C 2 ,a- estimate in the next subsection.

c

5.3. The C 2 ,a- estimate of cpo We now proceed to derive the 2 ,a_ estimate for cp, which by (2.50) and (2.54) is a solution to the complex Monge-Ampere equation: 0 log det ( gaiJ

+ CPaj3 )

= h

0 . + log det gaiJ =;= h,

and (2.77)

-

h(x, t)

= - f(x, t) + f(x, 0)

r

- -cp(x, t).

n By (2.70), there exists a constant A > 0 such that the Kahler metric gaiJ g~iJ + cp aj3 satisfies

~

(2.78) By the fact that the bounded function f satisfies t:::.f = R - r and that we have the scalar curvature bound (2.47), standard Lq theory (which only requires the uniform boundedness of the coefficient matrix (gaiJ) from above and below) implies that Ilfllco(M) + IIVV flILq(M) is bounded and hence by (2.77), Ilhllco(M) + IIVVhIILq(M) is bounded (uniformly in t when Cl (M) ::; 0) for any q < ;:0 (independent of t). Note that h is globally defined whereas h is only locally defined. However, for compactly contained open subsets U of a holomorphic coordinate chart of go, Ilhllco(u)+ IIVVhIlLq(u) and Ilhllco(u) + IIVVhIILq(U) are equivalent. Let B(R) denote the Euclidean ball of radius R centered at the origin in en. Since M is compact, there exists a finite collection of open sets {Uk}~~l and normal holomorphic coordinates Zk = {zn:=l defined on Uk (independent of t) such that No

B (3RkO) C Zk (Uk)

and

U z;;l (B (RkO)) = M, k=l

where RkO > O. Hence it suffices to prove the C2,a- estimate for cp in each open set z;;l (B (RkO)) assuming that (2.79)

Ilhllco(uk)

+ IIVVhIlLq(uk)

::; C

< 00.

From now on we work in a fixed coordinate chart. More precisely, we use Zk to push forward our discussion to the Euclidean ball B (3RkO)' For

90

2. KAHLER-RICCI FLOW

simplicity we drop the indices k in our notation below. Since 9 is Kahler, we may write 9Ct /3 locally as the complex (Hermitian) Hessian u Ct /3 of a function u. We shall show that the second derivative of u has bounded Holder norm. Now the equation reads locally as log det (u Ct /3) = h.

(2.80)

It is convenient to write log det as a function F(p), where p lies in the domain of positive definite Hermitian symmetric matrices. An important property that we shall make full use of is that F(p) = log det(p) is a concave function of p, a fact which can be easily checked. Taking the derivative of (2.80), we have

8F

- 8- uCt/3-y = h-y, PCt{3 82F

8

-8 - UCt/3-yUJ.Lii,,(

8F

+ -PCt(3 8_uCt/3-y"( = h-y"(

PCt{3 PJ.LV for each 'Y = 1, ... ,n. By the concavity of F we have (2.81)

On the other hand recall that 8F

--=U

a{3-

8Pa/3

=9

a{3-

,

where (uCt/3) is the inverse of (u a/3) = (9a/3).5 Therefore we can rewrite (2.81) as ~uw ~

(2.82)

h-y"(,

where

u-y"( and ~u denotes the Laplacian with respect to the metric 9Ct /3' For any R ~ Ro, let w

(2.83)

M(s)

~

= sup wand

m(s)

=

B(sR)

inf w. B(sR)

Also define the oscillation function:

w(sR)

~

M(s) - m(s).

The following weak form of the Harnack inequality, which holds in general for linear elliptic operators of divergence form, plays a crucial role in our estimate. One can find the proof of this result in various papers and 5T his is not different from the real version of this formula used in Volume One, which is the variation formula -logdetA = A -l)ij & Q &AQ

a

for any invertible matrix A ij .

(

a

5.

EXISTENCE AND CONVERGENCE

91

books on PDE, such as Moser [276], Morrey [274], Gilbarg and Trudinger [155], Han and Lin [195] (e.g., see Theorem 4.15 on p. 83 of [195]). THEOREM 2.61 (Harnack inequality). Let function such that

U

:

B(3Ro)

---+

lR be a C 2

1

Ao (tSa.a) ~ (ua.a) ~ Ao (tSa.a) forsomeA o E [1,00). Suppose that a nonnegative function v E W 2,2(B(3Ro)) and 9 E LQ(B(3Ro)), for some q > m/2, satisfy ~uv ~

9

in the weak sense in B(3Ro), where ~u denotes the Laplacian with respect to the metric ua.a. Then for any 0 < (j ~ T < 1 and 0 < p < m~2' there exists a constant C = C(p, q, 2n, Ao, (j, T) < 00 such that for any p ~ 2Ro, 1

(2.84)

(~r v (y)P dY) P ~ C ( inf v + i-rg-lIgIlLq(B(P») . pm JB(7'p) B(Op)

REMARK 2.62. The reason for why we can apply this theorem to (ua.a) = (ga.a) =

(g~.a+CPa.a)

is that the C 2 -estimate for cP yields (2.78). Note that

~u

(M(2) - w)

~

-h"(i and -h"(i E Lq for all q

~

00 (e.g.,

< 00), so we may apply the above theorem to M(2) -w (for example, with m = 2n, p = 2R,6 and (j = T = ~, so that (jp = Tp = R) to obtain for any q> nand 0 < p < that there exists C = C(p, q, n, A) < 00 such that q

n:1

1

r (M(2) - w (Y))PdY) P (R~ JB(R) n

~ C (M(2) -

(2.85)

M(1)

+ R 2 (q;n) Ilh"(iIILq(B(2R»)

since infB(R) (-w) = M(1). On the other hand, the concavity of F implies

F(Ui](X)) ~ F(Ui](Y))

8F

+ 8p .., (Ui] (y)) (Ui](X) -

Ui](Y))'

~J

Namely we have (2.86) The following linear algebra fact enables us to estimate w(R). 6Note that

R::; Ro.

92

2.

KAHLER-RICCI FLOW

LEMMA 2.63 (Linear algebra). There exist unitary vectors r}, . .. , rN E en with the property that, as Hermitian symmetric positive definite matrices, N

(giJ(y)) = Lav(y)rv®rv .

(2.87)

v=1

Here av(y) E lR and lA ~ av(y) ~ AA for some constant A > O. Moreover we may assume that the first n vectors r}, ... , rn form a unitary basis of en . REMARK 2.64. Let {ei}~=1 denote the standard basis for en and write rv ~ l:~=1 (rv)i ei for each 1/. By (2.87) we mean that N

(gil(y)) = L av(y) (rv)i (rv)1. v=1

EXERCISE 2.65. Prove the above linear algebra lemma. Define

Wv ~ Hess(u) (rv, rv) = uiJ(rvMrvk Now we let Mv(s) and mv(s) denote the quantities M(s) and m(s) defined by (2.83) using Wv instead of w. Then (2.86) implies N

L av(y)(wv(y) - wv(x)) ~ h(y) - h(x).

(2.88)

v=1

Choosing x E B(2R) to be a point where wdx) implies

= ml(2), this in particular

al(y)(wl(y) - Wl(X)) ~ h(y) - h(x) + L av(Y)(wv(x) - wv(y)) v~2

and hence

WI(y) - ml(2)

~ C(A, A) (RIIV'hllco + L(Mv(2) - wv(Y))) , v~2

where we used al(y) ~ Thus

lA' and av(y) ~ AA and wv(x) ~ Mv(2) for 1

{ (WI (Y) - ml(2))PdY) p (R; n iB(R)

~ C(A, A)RIIV'hllco 1

(2.89)

+ C(A, A) L (R;n v~2

( iB(R)

(Mv(2) - wv(y))PdY) P

1/

~ 2.

93

5. EXISTENCE AND CONVERGENCE

On the other hand, applying (2.85) to bound the V-norm of Mv(2) - WV , we have for each /J 2: 2, 1

r

(R; n JB(R) (Mv(2) - Wv (Y))PdY) (2.90)

:::; C ( Mv(2) - Mv(1)

P

+ R 2(q;n) II Hess(h) (rv, rv )IILq(B(2R))) .

Combining (2.89) and (2.90), we have

(2.91)

r (WI (y) - m1(2))PdY) (R; JB(R)

1

P

n

:::; C

(max (Mv(2) - Mv(l)) + RllVhllco + R 2(q;n) IIVVhIILq) . v:=::2

Now let

w(sR)

~ twv(SR) ~ tv=l (sup Wv v=l B(sR)

inf wv)

B(sR)

N

= L:(Mv(s) -mv(s)).

v=l We then have 1

r (WI(Y) - ml(2))p) P (R;n JB(R) (2.92)

:::; C (W(2R) - w(R)

+ RllVhllco + R 2(q;n) IIVVhIlLq)

On the other hand, from (2.85) we also have 1

r

(R; n JB(R) (Ml(2) - WI (y))p) P (2.93)

:::; C (W(2R) - w(R)

+ R 2(q;n) IIVVhIlLq)

.

.

94

2. KAHLER-RICCI FLOW

Putting these together, we obtain 1

wl(2R) = (

s; (

V01~(R) LIR) (Ml(2) - ml(2))p)'

V01~(R) LIR/W1 (Y) - ml(2))p

r 1

1

+ ( V01~(R)

LIR) (Ml(2) - (Y))P) , WI

~ C (W(2R) -

w(R)

+ RIIVhlico + R 2(q;n) IIVVhIILq) .

Since there is nothing special about the index 1, summing the corresponding upper bounds for w/I(2R) implies w(2R)

~ C (W(2R) -

with a different constant C

w(R)

+ RllVhllco + R 2 (q;n) IIVVhIILq)

< 00. We conclude the following.

LEMMA 2.66 (Oscillation estimate). There exists 6 < 1 (i. e., 6 = 1 such that lor any R ~ Ro we have on B(3Ro), (2.94)

w(R) ~ 6· w(2R) 2(q-n)

+ RllVhllco + R

2(q-n)

q

b)

_

IIVVhIILq.

_

Now since R q IIVVhIlLq(U) and IIVhllco(u) are bounded by (2.79), the Holder continuity of VVu on B(Ro) can be derived from (2.94) by a standard argument; see Moser [276], or Corollary 4.18 on p. 91 and Lemma 4.19 on p. 92 of Han and Lin [195], for example. Finally, the Holder continuity of VVu is equivalent to the Holder continuity of VV'P' 5.4. Proof of Theorem 2.51. Finally, we give the proof of Theorem 2.51, i.e., the proof of the convergence of the normalized Kahler-Ricci flow in the case where Cl < O. Assume, without loss of generality, that r = -n, which can be achieved by scaling the initial metric go. Notice that we have that I = - ~ satisfies

a

atl = b.g(t)1 - I and I/(x, t)1 ~ Ce- t and (2.48).) That is,

IVII (x, t)

~

Ce- t . (The last inequality is by

for some Cl < 00. 2 Now b.g(t) = gCt/3 az~az{3 and gCt{J = g~{J

+ 'PCt{J'

So the C2,Ct-estimate for

'P implies a CCt-estimate for the coefficients gCt{J. Thus we may apply the

6.

SURVEY OF SOME RESULTS FOR THE KAHLER-RICCI FLOW

95

parabolic Schauder estimate (e.g., Theorem 5 on p. 64 of Friedman [146]) to obtain Ilfll c 2 ,<>(M) ~ C2e- t for some C 2 < 00. Iterating the Schauder estimate, we have IlfII C 2m '<>(M) ~ Cme- t for some constants Cm < 00 and all mEN. This implies the estimate ~ C2 e- t and hence implies the exponential convergence of I ~ II C 2m ,<>(M)

'P (', t)

---t 'Poo (.) in Coo as t ---t 00 for some smooth function 'Poo. This proves that the normalized Kahler-Ricci flow converges in Coo to a Kahler-Einstein metric with negative scalar curvature. Theorem 2.51 is proved.

6. Survey of some results for the Kahler-Ricci How 6.1. Closed Kahler manifolds with nonnegative bisectional curvature. Using the short-time existence of the Kahler-Ricci flow, the result of Mori, Siu and Yau (Theorem 2.33) was generalized by Bando [19] when n = 3 and Mok [269] for n ~ 4.7 Mok also used techniques from algebraic geometry. THEOREM 2.67 (Kahler manifolds with nonnegative bisectional curvature). If (Mn,g) is a closed Kahler manifold with nonnegative bisectional

g)

curvature, then its universal cover (Mn, is isometrically biholomorphic to the product of complex Euclidean space, compact irreducible Hermitian symmetric spaces of rank at least 2, and complex projective spaces with Kahler metrics of nonnegative bisectional curvature. REMARK 2.68. Note that the above classification is up to isometry. Any complex projective space admits a metric with constant holomorphic sectional curvature (Le., the Fubini-Study metric). The proof of the theorem above uses the following result, proved by Bando for n = 3 and Mok for n > 4. We discuss this result further in Section 8 below. THEOREM 2.69 (KRF: nonnegative bisectional curvature is preserved). If(Mn,g(O)) is a closed Kahler manifold with nonnegative bisectional curvature, then the solution g (t) to the Kahler-Ricci flow has nonnegative bisectional curvature for all t ~ O. If in addition g (0) has positive Ricci curvature at one point, then g (t) has positive holomorphic sectional curvature and positive Ricci curvature for all t > O. Using the existence of a Kahler-Einstein metric, the convergence in the case of positive bisectional curvature was settled by Chen and Tian [87],

[88]. 7The case of nonnegative curvature operator was considered by eao and one of the authors [50].

96

2.

KAHLER-RICCI FLOW

THEOREM 2.70 (KRF: compact positive bisectional curvature). Suppose (Mn, 9 (0)) is a closed Kahler manifold with nonnegative bisectional curvature everywhere and positive bisectional curvature at a point. Then the solution 9 (t) to the normalized Kahler-Ricci flow, which has positive bisectional curvature for all t > 0, converges exponentially fast to the Fubini-Study metric of constant holomorphic sectional curvature on cpn. REMARK 2.71. Without using the existence of a Kahler-Einstein metric, Cao, Chen, and Zhu [49J proved a uniform curvature estimate (see Theorem 2.92). 6.2. Uniformization of noncom pact Kahler manifolds with nonnegative bisectional curvature. In this subsection we recall Yau's fundamental conjecture on the uniformization of complete noncompact Kahler manifolds with nonnegative bisectional curvature. CONJECTURE 2.72 (Noncompact Kahler uniformization Kc (V, W) > 0). If (Mn, g (0)) is a complete noncompact Kahler manifold with positive bisectional curvature, then M is biholomorphic to C n . Using the Kahler-Ricci flow on noncompact manifolds, Chau and Tam [65J proved the following result, which affirms Yau's conjecture in the case of bounded curvature and maximum volume growth. THEOREM 2.73 (KRF: noncompact positive bisectional curvature). If n (M , g (0)) is a complete noncompact Kahler manifold with bounded positive bisectional curvature and maximum volume growth, then M is biholomorphic to cn. There have been a number of works on the Kahler-Ricci flow on noncompact manifolds with positive bisectional curvature. For example, the reader may consult Shi [331], Tam and one of the authors [290], [292J, and Chen and Zhu [79], [80J. 6.3. Limiting behavior of the Kahler-Ricci flow on closed manifolds. There are also the following results about the limiting behavior of the Kahler-Ricci flow due to Sesum [323]. THEOREM 2.74. If (Mn, 9 (t)) , t E [0,00), is a solution to the KahlerRicci flow on a closed manifold with uniformly bounded Ricci curvature, then for any sequence ti -+ 00 there exists a subsequence such that (M, g (t + ti)) converges to (M~,goo (t)), where goo (t) is a solution to the Kahler-Ricci flow. The convergence is outside a set of real codimension 4. When n = 2, Sesum improved the above result to the following. THEOREM 2.75. If (M 2 ,g(t)) , t E [0,00), is a solution to the KahlerRicci flow on a closed manifold with uniformly bounded Ricci curvature, then for any sequence ti -+ 00 there exists a subsequence such that (M, 9 (t + ti)) converges to (M ~, goo (t)) , where goo (t) is a K ahler-Ricci soliton. The convergence is outside a finite number of points.

7. EXAMPLES OF KAHLER-RICCI SOLITONS

97

For some other recent work on the Kahler-Ricci flow the reader is referred to Phong-Sturm [305], [306], Chen [84J, Chen and Li [85], Song and Tian [337], Cascini and La Nave [60]' Tian and Zhang [351]' and [234J. 7. Examples of Kahler-Ricci solitons In this section, we provide a brief and regrettably incomplete sampling of some results on Kahler-Ricci solitons. These special solutions model singularities of the Kahler-Ricci flow. A Kahler-Ricci soliton is a Kahler manifold (Mn, g, J) such that the soliton structure equation

(2.95)

1

Rc+,\g + 2LXg

=

a

holds for some constant ,\ E IR and some real vector field X which is an infinitesimal automorphism (2.1) of the complex structure J. Note that X is an infinitesimal automorphism if and only if its (1, a)-part is holomorphic: a = '\7 o.X~ = a~'" Xf3. One imposes this requirement for the following reason. As we saw in Lemma 2.36, a solution of Ricci flow that starts with a Kahler metric on a complex manifold remains Kahler with respect to the same complex structure. On the other hand, if 'Pt is any family of diffeomorphisms of M, then each pullback 'Pt (g) is Kahler with respect to the complex structure 'Pt(J). Now consider the evolving metric h(t) := (1 + ,\t)'Ptg, where 'Pt is the family of diffeomorphisms generated by 2(1~At) X. If X is an infinitesimal automorphism of the complex structure, then 'Pt (J) = J, which implies that h( t) remains Kahler with respect to the same complex structure. Furthermore, using (2.95), it is easy to see that h solves the Kahler-Ricci flow:

gth = - Rc(h). One may also define a Kahler-Ricci soliton to be a Kahler manifold (M n , g, J) together with a constant ,\ E IR and a real vector field X satisfying the complex soliton equation (2.96) Equation (2.96) is equivalent to the conjunction of equation (2.95) and the statement that X is holomorphic. Notice that if we restrict our attention to gradient solitons (so that X is the gradient of a real-valued function), then (2.96) is equivalent to (2.95) without any extra hypotheses. (See §2.2 of [142J for the detailed argument.) 7.1. Existence and uniqueness. Any Kahler metric satisfying (2.96) with X = a is Kahler-Einstein. In this sense, Kahler-Einstein metrics may be regarded as trivial Kahler-Ricci solitons. 8 So if no Kahler-Einstein metric exists, a natural replacement is a Kahler-Ricci soliton. In fact, existence of a Kahler-Einstein metric and a nontrivial gradient Kahler-Ricci soliton 80f course, there is nothing 'trivial' about Kii.hl.er-Einstein metrics!

98

2.

KAHLER-RICCI FLOW

are mutually exclusive: applying the Futaki functional F[wJ to the holomorphic vector field X = grad f, one gets

Moreover, Kahler-Ricci solitons on a compact Kahler manifold (Mn, J) are unique up to holomorphic automorphisms. (See Tian and Zhu [352, 353, 354J.) Specifically, we have the following theorem. THEOREM 2.76 (Uniqueness of Kahler-Ricci solitons). Let (Mn, J) be a compact Kahler manifold. If metrics 9 and 9' on M satisfy (2.95) with respect to holomorphic vector fields X and X', respectively, then there is an element CT in the identity component of the holomorphic automorphism group such that 9 = CT*g' and X = (CT-I)*X'. For recent results on uniqueness and other properties of noncom pact Kahler-Ricci solitons, see [63, 64J, [35J and [78J. One might ask, therefore, whether there exists either a Kahler-Einstein metric or else a Kahler-Ricci soliton on every compact Kahler manifold Mn with CI (M) > O. The answer is yes if n ~ 2. A compact complex surface with CI > 0 is p2#k p2 for some k E {O, 1, ... , 8}. (Here and below, pn = cpn is complex projective space.) A Kahler-Einstein metric exists for k = 0 and 3 ~ k ~ 8. (See Theorem 2.34.) In the remaining cases k = 1,2, there is a (non-Einstein) Kahler-Ricci soliton. (See [239], [366], [47], and Section 7.2 below.) In higher dimensions, however, the answer is no. There exist 3-dimensional compact complex manifolds that admit no Kahler-Einstein metric and no holomorphic vector fields, hence no KahlerRicci soliton structure. (See [346, §7J as well as [215, 216J and [278J.) More generally, Tian and Zhu have exhibited a holomorphic invariant that generalizes the Futaki invariant and acts as an obstruction to the existence of a Kahler-Ricci soliton metric [354J on a compact complex manifold (Mn, J). 7.2. The Koiso solitons. As was noted in Proposition 1.13 or Proposition A.32, all compact steady or expanding solitons are Einstein. This is not true for shrinking solitons. The first examples of nontrivial (Le. nonEinstein) compact shrinking solitons were discovered by Koiso [239J and independently by Cao [47J. These are Kahler metrics on certain k-twisted projective-line bundles pI ~ Fk -* pn-I first described by Calabi [44J. We will discuss their construction in considerable detail, because it serves as a prototype for later examples. We begin with Calabi's bundle construction. pn-I is covered by n charts (CPa: Ua ~ cn-I), whereUa = {[XI, ... ,XnJ E pn-I: Xa =F O} Xa-l Xa±l ~) • (lXT . an d CPa.. [Xl"",Xn J f---t (~ , ••• , - - , , ••• , vve wn't e Xa InXa: Xo; Xo: Xo. a stead of x here and in the next paragraph in order to simplify some formulas below.) In particular, one may define complex projective space by

7. EXAMPLES OF KAHLER-RICCI SOLITONS

99

lpm-l = (Il~=ICPo(Uo))/~, where, for example,

CPI (UI ) :3 (Zl, ... , Zn-l )

~

1 Z2 Zn-l) ( -, -, ... , - E CP2 (U2). Zl Zl Zl

Given kEN, we formally identify pI = e u {oo} and define the k-twisted bundle :Fk = (Il~=l (Uo X pI)) / "', where Uo x pI :3 ([Xl, .. . , Xnl;~) '" ([YI, .. . , Ynl; 1]) E U(3 X pI if and only if [Xl' ... ' xnl = [YI, ... , Ynl and 1] = (~)k~ for each Q. Equivalently, one may define where, for example, 1 Z2 Zn-l k I (-, -, ... , - ; Zl () E CP2(U2) x p . Zl Zl Zl Notice that 80 = {[Xl, ... ,xnl; o} and 8 00 = {[Xl, ... , xnl; oo} are two global sections of Fk. The key to constructing Kahler-Ricci solitons on F'J: (as well as examples on other topologies to be considered below) will be to find a Kahler potential on en \ {o} satisfying certain symmetries and boundary conditions. To see = Fk\(80 U 8 00 ) and define 'IjJ : en\{O} - t so that why this is so, let

CPI(UI ) x p

I

:3 (Zl, ... , Zn-l;

()

~

Fr:

Fr:

'IjJ: (XI, ... ,Xn ) f---+ ([xI, ... ,xnl;x~) if Xo =1= O. It is easy to see that ([Xl, ... , xnl; X~) '" ([Xl, ... , xnl; x~) whenever Xo =1= 0 and x(3 =1= 0, hence that 'IjJ is well defined. The map 'IjJ is clearly surjective. If 'IjJ(XI, ... ,xn) = 'IjJ(YI, ... ,Yn), where, say Xo =1= 0 and Y(3 =1= 0, then

(Xl, ... , Xa-l , Xa+l , ... , Xn; X~) ~ (YI , ... , Y(3-1 , Y(3+1 , ... , Yn; Y~). Xa Xa Xa Xa Y(3 Y(3 Y(3 Y(3 The equivalence relation ~ then implies that Y~ = x~, hence that Y(3 = (}x(3 for some k-th root of unity (). Because [Xl, ... , xnl = [YI, ... , Yn], it follows that Y'Y = (}x'Y for all T = 1, ... , n, hence that 'IjJ is a k-to-one map. Therefore, a Kahler potential P on e n \ {O} will induce a well-defined Kahler metric on provided that 8ap((}xI, . .. , (}x n ) = 8ap(XI, ... , xn). With these considerations in mind, our method will be to construct a suitable Kahler potential P : en \ {O} - t IR whose asymptotics as Izi - t 0 and Izi - t 00 ensure that the induced metric extends smoothly to 8 0 and 8 00 • If we are interested in shrinking solitons, what properties should P possess? Let's suppose that P determines a Kahler metric g. As above, we take the Kahler and Ricci forms to be w = Agai3 dz o 1\ dz(3 and p = A Rai3 dz a 1\ dz(3, respectively. Then (locally) we have

Fr:

82

g-a(3 - 8za8z(3 P

2. KAHLER-RICCI FLOW

100

and

82

= - 8z Ci 8z!3 log det g,

RCi[J

exactly as in (2.6). If Q denotes the soliton potential function with gradient vector field X, then equation (2.96) reduces to

82

8zCi 8z!3 (log det 9 - Q - AP) = O. Because we are interested in shrinking solitons, we may assume that A < O. Then (modifying the Kahler potential P by an element in the kernel of V'V if necessary) we may assume that Q = log det 9 - AP. This will give us a soliton provided X = grad Q is holomorphic, that is, provided that

o = ~X!3 =~ ( !3'Y~Q) . 8zCi 8zCi 9 8z'Y Substituting Q = log det 9 - AP, we obtain a single fourth-order equation for the scalar function P, namely (2.97)

8~Ci

[gf3'Y

8~'Y (log det 9 -

AP)]

= O.

To proceed, we adopt the Ansatz that the potential P is invariant under the natural U(n) action in the sense that it is a function of r = log I:~=1 IzCi l2 alone. In this case, setting cP = Pr, we have (2.98)

gCi[J

= e- rcp8Ci!3 + e- 2r (CPr - cp)ZCi Z!3,

so our P will be a Kahler potential if and only if cP and CPr are everywhere positive. Now (following [47] and [142]) we can write (2.97) as the fourthorder ODE n n n Prrrr - 2P;rr T + nrrrr - ( n - 1) P;r p2 + 1\'( rrrrrr - p2rr ) = O. rr r We shall see that only two of the four arbitrary constants in its solution are geometrically significant. Q = ~ZCi will be holomorTo simplify (2.99), notice that XCi = gCi[J~~ z rrr phic if and only if Qr = p,Prr for some p, E lR. Since 9 will be Kahler-Einstein if X = 0, we may assume that p, i= O. Substituting Q = log det 9 - AP, one then obtains

( ) 2.99

(logCPr)r + (n -l)(logCP)r - P,CPr - Acp - n

= 0,

which is a second-order equation for cP, hence a third-order equation for P. (This integration can also be accomplished by standard ODE techniques.) Because CPr > 0 everywhere, one may regard r as a function of cP and hence F(cp). One finds (remarkably) that F satisfies a linear may write CPr equation

n-l

F' + (-cP

-

p,)F - (n+ Acp)

= 0,

7.

EXAMPLES OF KAHLER-RICCI SOLITONS

101

whose solution is

r.pr = r.pl-neJ.l.'P(1/ + >..In + nIn-l), where

1/

is another arbitrary constant and In

r.pr = I/r.p

I-n

= J r.pne-J.l.'P dr.p. This leads to

J.I.'P >.. >.. + /1. ~ n! j"J+1-n e - -r.p - l+n L..J -., /1. 'I-' , /1.

/1.

j=O

J.

which is a separable first-order equation for r.p. The third and fourth arbitrary constants arise when one finds the implicit solution r( r.p) and then integrates r.p to obtain P. These are geometrically insignificant, because they disappear in (2.97); the two geometric degrees of freedom are the parameters /1. i= 0 and 1/. In summary, what we have thus far accomplished is to construct a potential function for a U(n)-invariant Kahler-Ricci soliton metric (2.98) on Cn\{O}, hence a possibly incomplete Kahler-Ricci soliton metric on What remains is to choose /1. and 1/ in order to get a complete metric on FJ:. Our choices of the two parameters will be determined by the two boundary conditions as r -4 ±oo. Since r.pr > 0, we may define a < b E [0,00] by a = limr---+_oor.p(r) and b = limr-+oo r.p(r). If a > 0, one may write P(r) = ar + p(e Qr ) in a neighborhood of Izl = 0, with p smooth at zero, p(O) = 0, p'(O) > O. Similarly, if b < 00, one may write P(r) = br + q(e-f3r ) in a neighborhood of Izl = 00, with q smooth at zero, q(O) = 0, q'(O) > O. One then takes advantage of the following observation.

Jr.

LEMMA 2.77 (Calabi). Assume that k > 0 and a, bE (0,00). (1) When a = k, the potential P(r) = ar + p(e Qr ) induces a smooth Kahler metric on a neighborhood of So in FJ:. Any pI in So has area a7r. (2) When (3 = k, the potential P(r) = br + q(e- f3r ) induces a smooth Kahler metric on a neighborhood of Soo in FJ:. Any pI in Soo has area b7r. For the proof, see [44] or [142, Lemma 4.2]. With more work, one finds that it is possible to satisfy both boundary conditions by appropriate choices of /1. and 1/. (See [47] or adapt the arguments in [142, §4.1].) Normalizing by fixing>.. = -1, these choices yield a = n - k and b = n + k. Since one needs a > 0, one obtains a unique gradient shrinking Kahler-Ricci soliton on FJ: for each k = 1, ... , n - 1. These are the Koiso solitons. 7.3. Other U(n)-invariant solitons. The construction we have described in Section 7.2 above has natural generalizations allowing the discovery of other explicit Kahler-Ricci soliton examples. Namely, one searches the 2-dimensional (/1., 1/) parameter space of U (n)- invariant Kahler potentials

102

2. KAHLER-RICCI FLOW

P(r) on en\{O} for those whose behavior at the boundary izl = 0 implies that 1_: the metric is completed by adding a smooth point at r = -00; 2_: the metric is completed by adding an orbifold point at r = -00; 3_: the metric is completed by adding a Ipm-l at r = -00; or 4_: the metric is complete as r - t -00; and whose behavior at the boundary Izl = 00 implies that 1+: the metric is completed by adding a smooth point at r = +00; 2+: the metric is completed by adding an orbifold point at r = +00; 3+: the metric is completed by adding a pn-l at r = +00; or 4+: the metric is complete as r - t +00. Of course, not all combinations of these alternatives are globally compatible. For example, it is easy to see that the growth condition cpr > 0 prohibits completing the metric by adding a Ipm-l at Izl = 0 and a smooth point at Izl = 00. Nonetheless, this has been a productive line of research. In the remainder of this section, we will survey some of its results. It is possible to add a smooth point at Izl = 0 and to construct a unique steady Kahler-Ricci soliton on en that is complete as Izl - t 00. In complex dimension n = 1, this is just the cigar soliton discovered by Hamilton and discussed in Chapter 2 of Volume One; the examples in higher dimensions are due to Cao [47]. These solitons have the following asymptotic behavior: in the sphere s2n-1 at metric distance r » 0 from Izl = 0, the Hopf fibers U(1) . z have diameter 0(1), while the pn-l direction has diameter O(Vr). Accordingly, one calls this cigar-paraboloid behavior. It is also possible to add a smooth point at Izl = 0 and to construct expanding Kahler-Ricci solitons on en that are complete as Izl - t 00. There is in fact a 1-parameter family (en,go)o>o of such examples in each dimension, due to Cao [48]. (It is a heuristic principle that expanding solitons are easier to find than their shrinking cousins. For these examples, satisfying the boundary condition at Izl = 0 reduces the parameter space by one dimension, but completion as Izl - t 00 comes for free.) Each soliton (en, go) is asymptotic as Iz I - t 00 to the Kahler cone (en \ {O}, go), where the metric go is induced by the Kahler potential P(r) = eOr jB. For each k = 2,3, ... , the authors of [142] add an orbifold point at Izl = 0 and a Ipm-l at Izl = 00 to construct a unique shrinking Kahler-Ricci soliton on an orbifold, which is called Yk' The compact orbifold Yk may be regarded as Ipm jZk branched over the origin and the pn-l at infinity. The orbifold singularity at the origin is modeled on en jZk. For each dimension n ~ 2 and k = 1, ... , n -1, the authors of [142] add a k-twisted pn-l at Izl = 0 and construct a unique shrinking Kahler-Ricci soliton metric that is complete as Izl - t 00. The resulting soliton has the topology of the complex line bundle e ~ L"!:..k - pn-l characterized by (CI' [E]) = -k, where CI is the first Chern class of the bundle and E ~ pI is a positively-oriented generator of H2(pn-l; Z). (For example, the total

8.

KAHLER-RICCI FLOW WITH NONNEGATIVE BISECTIONAL CURVATURE

103

space of L"!:..I is simply Cn blown up at the origin.) As Izl --t 00, the soliton metric is asymptotic to a Kahler cone (C n\{0},90)/Zk, where 0 = O(n,k). For each dimension n 2: 2, Cao [47] adds an n-twisted IF-I at Izl = 0 and constructs a unique complete steady Kahler-Ricci soliton on the total space of the bundle C '---t L"!:..n - pn-I. The metric exhibits cigar-paraboloid behavior at infinity. For each dimension n 2: 2 and k = n + 1, ... , the authors of [142] add a pn-I at Izl = 0 and construct a I-parameter family of complete expanding Kahler-Ricci solitons on C '---t L"!:..k - IF-I. The solutions are parameterized by 0 > 0, where (cn\{0},90)/Zk is the asymptotic Kahler cone at infinity.

8. Kahler-Ricci flow with nonnegative bisectional curvature The study of the Kahler-Ricci flow of Kahler metrics with nonnegative bisectional curvature is somewhat analogous to the study of the Riemannian Ricci flow of metrics with nonnegative curvature operator. One aim is to uniformize Kahler metrics with nonnegative bisectional curvature in both the compact and noncompact setting. In particular, one would like to flow such metrics to canonical metrics, or to infer the existence of canonical metrics from the long-time behavior of the flow. One would also like to deduce properties of the underlying complex structure of the Kahler manifold, and when possible, classify the manifold up to biholomorphism.

8.1. Nonnegative bisectional curvature is preserved. Consider the Kahler-Ricci flow 9a-'!3 = -RafJ. We shall prove that the Kahler-Ricci flow preserves the nonnegativity of the bisectional curvature. As in the real case, the key is Hamilton's weak maximum principle for tensors. This result was proved first by Bando for n S 3 and by Mok in any dimension. (See Theorem 2.69 above.) The result was also extended to the complete noncompact case by W.-X. Shi under the additional assumption of the bisectional curvature being bounded. We say that a Kahler metric has quasi-positive Ricci curvature if the Ricci curvature is nonnegative everywhere and positive at some point.

gt

THEOREM 2.78 (Nonnegative bisectional curvature preserved). The nonnegativity of the bisectional curvature is preserved under the Kahler-Ricci flow on closed Kahler manifolds. Moreover, if the initial metric also has quasi-positive Ricci curvature, then both the Ricci curvature and the holomorphic sectional curvature are positive for metrics at positive time. The basic computation in the proof of the above result is the following.

104

2.

KAHLER-RICCI FLOW

PROPOSITION 2.79 (Evolution equation for the curvature). Under the Kahler-Ricci flow, (2.100)

(:t - ~)

Ro.iJrJ

= Ro.jiIlJ R J.1.i3'""(V -

Ro.ji'""(vR J.1.i3I1J

+ Ro.i3l1ji R J.1.v'""(J

1

- "2 (Ro.ji R J.1.i3'""(J + RJ.1.i3 R o.ji'""(J + ~jiRo.i3J.1.J + RJ.1.J R o.i3'""(ji) . REMARK 2.80. The Riemannian analogue of this formula is given by Lemma 6.15 on p. 179 of [108].

In the proof of the proposition we find it convenient to use a formula relating ordinary derivatives and covariant derivatives at the center of normal holomorphic coordinates. LEMMA 2.81 (Relation between ordinary and covariant derivatives). If 'fJ is a closed (1, I)-form, then, at the center of normal holomorphic coordi-

nates, we have

(2.101)

{p

\7 i3 \7 0. 'fJ'""(J

=

{)zo.{)z{3 'fJ'""(J + 'fJ)..J R o.i3'""()'" {)2

(2.102)

\7 0. \7 i3'fJ'""(J

=

{)zo.{)z{3 'fJ'""(J + 'fJ'""().,Ro.i3)..J.

PROOF. We compute that at the center of normal holomorphic coord inates, \7i3 \7 0. 'fJ'""(J = {)i3 \1 0. 'fJ'""(J - r~o \7 0. 'fJ,""(e

= {)i3 ({)o.'fJ'""(J - r;'""('fJgJ) = {)i3{)o.'fJ'""(J - {)i3 r ;'""('fJgJ {)2

{)zo.{)z{3 'fJ'""(J

+ R~i3'""('fJgJ'

where we used (2.4) in the last line; this proves (2.101). Note that (2.102) 0 is just the conjugate of (2.101). Now we give the PROOF OF PROPOSITION 2.79. We compute the evolution equation for Ro.i3'""(J at any point x and time t using normal holomorphic coordinates {zo.} centered at x with respect to 9 (t) . In such coordinates, a;;~ (x, t) = Recall from (2.5) that R ____ {)2go.i3 o.{3'""(O {) z'""( {)ZO

+ 9

pu {)go.u {)gpi3 {)z'""( {) ZO .

o.

8. KAHLER-RICCI FLOW WITH NONNEGATIVE BISECTIONAL CURVATURE 105

This implies

8

8 (8) 8tga~

= 'V"I 'V &Ra~ -

(2.103)

8 2

2

8t Raih & = - 8z"l8z0

=

8Z"l8zoRa~

Ra>..R"I&>"~'

where we used (2.102). Since (2.103) is tensorial, it holds in any holomorphic coordinate system. We wish to compare the above formula with t:.Ra~"I& ~

1

2" ('V /1- 'V jl + 'V jl 'V /1-) Ra~"I&'

To this end we compute (apply the second Bianchi identity (2.9) and commute covariant derivatives (Exercise 2.22)) 'V"I'VJRa~ = 'V"I'V&Ra~/1-jl = 'V"I'VjlRa~/1-&

= 'V jl 'V /1-Ra~"I& - ~jl/1-iJRa~v&

~jlaiJRv~/1-&

+ R"Ijlv~RaiJ/1-&

+ R"Ijlv&Ra~/1-iJ

and

'V jl 'V /1-Ra~"I& = 'V /1- 'V jlRa~"I& + R/1-jlaiJRv~"I& - R/1-jlv~RaiJ"I&

+ R/1-jl"liJRa~v& - R/1-jlv&Ra~"IiJ = 'V /1- 'V jlRa~"I& + RaiJRv~"I& + R"IiJRa~vJ - Rv&Ra~"IiJ'

Rv~RaiJ"I&

Combining the formulas above yields

8 8t Ra~"I& = 'V jl 'V /1-Ra~"I&

- ~jlaiJRv~/1-&

+ R"Ijlv~RaiJ/1-&

+ R"IjlvJRa~/1-iJ - Ra>..R"I&>"~ = t:.Ra~"I& - ~jlaiJRv~/1-& + R"Ijlv~RaiJ/1-& + R"IjlvJRa~/1-iJ - ~iJRa~v&

1

- 2" (RaiJRv~"IJ + Rv~RaiJ"(& + ~iJRa~v& + Rv&Ra~"IiJ)

,

o

and the proposition follows.

As a consequence of the proposition we have the following evolution equations for the bisectional curvature and the Ricci tensor. COROLLARY

(! -

2.82 (Bisectional curvature evolution).

t:. ) RaOt"l'Y =

t

(IRajlv'Y12 - IR ajl"liJl2

+ RaOtvjlR/1-iJ"I'Y)

/1-,v=l n

(2.104)

- L Re (RajlR/1-Ot"l'Y + ~jlRaOt/1-'Y) . /1-=1

Here Re(A) = ~(A +..4) denotes the real part of a complex number A.

2. KAHLER-RICCI FLOW

106

PROOF.

Indeed, substituting j3

= a and 8 = l' in (2.100), we have

(%t - A ) Rcxory'Y = RCXilv'YRJ.l.c,'Yv - RCXil'YvRJ.l.c,v'Y

+ Rcxc,vilRJ.l.v'Y'Y

1

- .2 (RcxilRJ.l.c,'Y'Y + RJ.l.c,Rcxil'Y'Y + ~ilRcxc,J.I.'Y + RJ.I.'YRcxc,'Yil) . o COROLLARY 2.83 (Ricci tensor evolution). The Ricci tensor satisfies the Lichnerowicz heat equation:

a

at RcxfJ = ARcxfJ + RcxfJ'YiS R 6'Y -

(2.105)

Rcx'YR'YfJ

= ALRcxfJ·

REMARK 2.84. More generally, we say that a real (1, I)-tensor hcxfJ satisfies the Lichnerowicz heat equation if a 1 1 athcxfJ = ALhcxfJ ~ AhcxfJ + RcxfJ'YiS h6'Y - .2 Rcx'Yh'YfJ - .2 R 'YfJ hcx'Y'

See also (2.22). PROOF.

Summing (2.100) over 'Y = 8 from 1 to n, we have

(%t - A ) Rcxi3 =

t

(%t - A ) Rcxi3'Y'Y

+ R6'YRcxfJ'YiS

k=l

= RCXilV'YRJ.l.fJ'Yv -

RCXil'YvRJ.l.fJv'Y

+ Rcxi3vilRp.v + RcxfJ'YiSR6'Y

1

- .2 (RCXil R p.i3 + Rp.i3 Rcxil + ~ilRcxfJp.'Y + Rp.'YRcxfJ'Yil) = RcxfJ'YiS R 6'Y - R cxil RJ.l.i3'

o

after cancelling terms to get the last equality.

REMARK 2.85. Equation (2.105) may also be derived from (2.36), (2.10) and commuting covariant derivatives. In particular, 1 ARcxi3 = .2 (V'Y V'Y + V'Y v\) Rcxi3

1

=

1

.2 V'Y V fJRcx'Y + .2 V'Y V cxR'YfJ 1

1

= V cx V fJR - .2 R'YfJcxiS R 6'Y + .2 R6fJ R cxiS 1

(2.106) That is,

1

+ .2 RcxiS R 6i3 - .2 R'Ycxj36 R 'YiS = V cx V i3 R - RcxfJ'YiSR6'Y + RcxiSR6fJ. V cx V fJR = ALRcxfJ·

Based on the evolution equation (2.104) and Hamilton's maximum principle for tensors (see Chapter 4 of Volume One or Part II of this volume) we present the following.

8.

KAHLER-RICCI FLOW WITH NONNEGATIVE BISECTIONAL CURVATURE

107

PROOF OF THEOREM 2.78. (I) We first prove that the nonnegativity of the bisectional curvature is preserved under the flow. Analogous to Theorem 4.6 on p. 97 of Volume One, by the Kahler version of the maximum principle for tensors (see Proposition 1 in §4 of [19]), we need to show that the quadratic on the RHS of (2.104), i.e., (2.107)

QOdi'Y"Y

~

n

L (IRctjLlI"Y1 p.,I1=l

2 -IRctjL")'iiI 2

+ RctiilljLRp.ii")'''Y)

n

- L Re (RctjLRp.ii")'''Y + ~jLRctiiP."Y) , p.=1

satisfies the null eigenvector assumption. That is, we assume Rctii,,),"Y = 0 for some a and 'Y at some point x, and we shall prove that (2.108) at x. First observe that since Rctii'Y"Y = 0 at x and the bisectional curvatures are nonnegative, we have at x, n

n

L RctjLRp.ii")'''Y = p.=1 L ~jLRctiip."Y = o.

p.=1

By (2.107), in order to prove (2.108) at x, it suffices to show that n

n

L RctiilljLRp.ii,,),"Y ~ p.,I1=l L (I R ctjL")'iiI p.,I1=l

(2.109)

2

-IRctjLlI"Y1 2 ) .

We shall prove (2.109) below, but first we show how the positivity of the Ricci tensor and holomorphic sectional curvatures follow from the quasipositivity of the Ricci curvature at t = O. (2) Recall that the Ricci tensor Rct/3 satisfies the Lichnerowicz heat equation, so that by taking a = {3 in (2.105), we have

a

at R ctii =

(2.110)

b.Rctii

+ Rctii'Y8 R 8"Y -

Rct"Y~ii'

Since the nonnegativity of the bisectional curvature is preserved, by applying Hamilton's strong maximum principle for tensors to (2.110) (see Theorem A.53 and also Part II of this volume), the Ricci tensor becomes positive for all positive time. Now suppose there exists a space-time point (xo, to), with to > 0, at which some holomorphic sectional curvature R ctiictii is zero. Since the holomorphic sectional curvature is nonnegative everywhere, by (2.104), we have at (xo, to),

o~

(:t - b.)

R ctiictii

=

t

p.,I1=l

(2 1R ctiilljL 12 - IRctjLctiil 2) ,

2. KAHLER-RICCI FLOW

108

where we used I::=1 Re (Rap.R/-,aaa + Rap.Raa/-,a) other hand, by (2.109) with Q = I, we have n

=

a at (xo, to). On the

n

L

L

IR aavp.12 ~

IR ap.ailI 2 .

/-"v=l

Hence we conclude Raavp. = RaP.ail = a at (xo, to) for all j.L, v. This in turn implies Raa = a, which is a contradiction. (1) continued. We now verify (2.109). Consider the following Hermitian symmetric form defined by the bisectional curvatures: (-a a +sX'-a a +sX'-a a +sY'-a a +sY ) ~a Q(X,y,s)~Rme za

~

~

~

for X, Y E r 1,0 M and s E R At a point where the bisectional curvatures are nonnegative and Raa'Yi = a, we have Q(X, Y, s) ~ a and Q(X, Y, a) = a for all X, Y E 1,0 M and s E R Therefore the second variation at s = a is nonnegative:

r

d2

a:::; ds 2

1

= Rme

_

8=0

Q(X, Y, s)

( -a8) + X,X, az'Y' az'Y

+ 2 Re ( Rme

( X,

Rme

a~a ' Y, a~'Y )

(a8-) Y, Y aza' aza'

+ Rme ( X,

a~a' a~'Y' Y) )

.

In terms of a unitary (1, a)-frame {ei}i=l' we may write this as Ri3'YiXi Xj

+ 2 Re ( RiajiXiyj + Ria'Y3XiYj) + Raai3 yiyj ~ a,

where X ~ I:~1 Xiei and Y ~

I:j=l yjej. By Lemma 2.86 below, we have

n

n

L Ri3'YiRaaj'i ~ L (l~ajiI2 -I Ria'Y31 2) . i,j=l i,j=l The claimed inequality (2.109) follows and this completes the proof of Theorem 2.78. D LEMMA 2.86. Let Q(X, Y) be a Hermitian symmetric quadratic form

defined by Q(X, Y) = Ai3Xi Xj

+ 2 Re ( BijXiyj + Di3XiYj) + Gi] yiyj.

If Q is semi-positive definite, then n

L

i,j=l

Ai3 Gj'i

~

n

L

i,j=l

(I B ij12 -I Di312).

9.

MATRIX DIFFERENTIAL HARNACK ESTIMATE

109

For the proof of this lemma, which is elementary in nature, we refer the reader to Mok [269]. 9. Matrix differential Harnack estimate for the Kahler-Ricci flow In this and the next section we discuss various differential Harnack estimates for the Kahler-Ricci flow and their geometric applications. Differential Harnack estimates for the Riemannian Ricci flow will be discussed in Part II. For Kahler-Ricci flow, a fundamental result is H.-D. Cao's differential Harnack estimate for solutions with nonnegative bisectional curvature (see [46]). Define (2.111) a J RaP Z {X)aP ~ at RaP + RaiRyP + '\l'YRaPX'Y + '\l iRaPX'Y + RaP'YJX'Y X + -tfor any {l,O)-vector X

= X'Y 8~"" and where Xi ~ X'Y.

THEOREM 2.87 (Kahler matrix Harnack estimate). If (Mn,g{t)) is a complete solution to the Kahler-Ricci flow with bounded nonnegative bisectional curvature, then (2.112) for any {I, O)-vector X.

This result may be considered as the space-time analogue of Theorem 2.78. We shall also see a similar analogy for the Riemannian Ricci flow in Part II, where Hamilton's matrix differential Harnack estimate will appear as the space-time analogue of the result that nonnegative curvature operator is preserved under the Ricci flow. 9.1. Trace differential Harnack estimate for the Kahler-Ricci flow. Taking the trace of the estimate (2.112) leads to the so-called trace differential Harnack estimate, after applying the second Bianchi identity. COROLLARY 2.88 (The Kahler trace differential Harnack estimate). Let (Mn,g{t)) be a complete solution to the Kahler-Ricci flow with bounded nonnegative bisectional curvature. Then (2.113) PROOF. By (2.112) we have O ~ 9apZa(3- -- 9ap

R ata R a(3- + 1R a(3_1 2 + ~v 'Y RX'Y + ~v i RXi + R'YfJ-X'YXJ + t'

and (2.113) follows from (2.37).

0

2. KAHLER-RICCI FLOW

110

> 0, the {l,O)-vector minimizing the LHS of (2.113) is X'Y = I - (Rc- )'YP\7 pR, where (Rc-ItP R'YP = t5~. Hence, ifRc > 0, then (2.113) When Rc

is equivalent to

aR at

+ Rt

_ (Rc- I )'Y5 \7 R\7-R > O. 'Y

r

Since R'Y5 ~ Rg'Y5' we have - (Rc- I 5 ~ {}

(2.114)

at log (tR) - 1\7 log (tR)1

2

6-

_~g'Y5, {}

and hence

1

= at log R + t

- 1\7 log RI 2 ~ o.

Without assuming Rc > 0, we still obtain (2.114) by taking X'Y = -~\7'YR in (2.113) and using RaP ~ R9ap . COROLLARY 2.89

(Integrated form of Kahler trace Harnack estimate).

If (Mn,g(t)) is a complete solution to the Kahler-Ricci flow with bounded nonnegative bisectional curvature, then for any Xl, X2 E M and 0 < II < t2, we have R (X2' t2) > -tl e _l~ ----'----'4 R(XI,tl)-t2 ' where ~ = ~ (Xl, tl; X2, t2) ~ inf'Y fttl2 Ii' (t)I;(t) dt, and the infimum is taken over all paths 'Y : [tl, t2J ~ M with 'Y (tl) = Xl and 'Y (t2) = x2· By the fundamental theorem of calculus and (2.114), we have for any'Y : [tl, t2J ~ M with 'Y (tI) = Xl and 'Y (t2) = X2, PROOF.

log

t2R (X2' t2) lt2 d R( ) = -d [log tR b (t) , t) J dt tl

XI. tl

t

tl

= 1lt2 [(:tlogtR) ('Y(t),t) + (\7logtR,i'(t))g(t)] dt

~ lt2

[1\7l0gtRI;(t)

tl

+ (\7l0gtR,i'(t))g(t)]

dt

~ -~ lt21i' (t)I;(t) dt.

tl The corollary follows from taking the infimum over all 'Y.

o

For the normalized Kahler-Ricci flow, we have the following. COROLLARY 2.90 (Integrated trace Harnack for normalized Kahler-Ricci flow). If (Mn,g(t)) is a solution to the normalized Kahler-Ricci flow on a closed manifold with nonnegative bisectional curvature, then for any Xl, X2 E M and 0 < tl < t2,

(2.115)

where

~

is as above and r

~

0 is the average (complex) scalar curvature.

> eii-tl-l R EMARK 2 .91 . Note t h at l-e-ii-tl l-e-nrt 2 - ~. en 2-1

111

9. MATRIX DIFFERENTIAL HARNACK ESTIMATE

Before we prove the corollary, we first recall how to go from the KahlerRicci flow to the normalized Kahler-Ricci flow on closed manifolds. Let 9 (t) be a solution to ~gQJ3 = -ROti3 and let 9 (t) ~ 'IjJ (t) 9 (t) , where 'IjJ (t) is to be defined below. We compute

_.

{)

{)t9Oti3 = -'ljJROti3 since ROti3 we have

+ 'ljJgOti3

= ROti3' Hence if we define a new time parameter i by -Nt = i %t'

{) -

{)igOti3

= -

~-

R

Oti3 + 'ljJ2 gOti3'

In particular, to obtain the normalized Kahler-Ricci flow, where mains in the same Kahler class, we set ~ (t)

f

'IjJ (t)2

n

9 (t) re-

,

where f is the average scalar curvature of 9 (t), which is independent of time since 9 (t) stays in the same Kahler class. Thus we take 'IjJ (t) ~ (1 - ~t) -1 . Since di = 'ljJdt, we may take i ~ -~ log (1- ~t) . That is, t

(1 - e-~l) .

=~

2.90. Let 9 (i) be a solution of the normalized Kahler-Ricci flow %t 9Ot i3 = -ROti3 + ~9Oti3' Then PROOF OF COROLLARY

is a solution of the Kahler-Ricci flow and we have the estimate ~~;;::;~ >

~e-~A. This implies

where

Li (Xl, i I ; X2, i2) = inflt21 ddT' "f

since

tl

t

(t)1

2

g(t)

dt

= inf "f

ft = 'IjJ-9t, 9 (t) = 'IjJ-I 9 (i) , and dt = 'IjJ-Idi.

(21

itl

d~ (i) 12 dt

ii(i)

di

o

Since 9 (t) has nonnegative bisectional curvature, and in particular it has nonnegative Ricci curvature, under the normalized Kahler-Ricci flow we have itgOti3 ~ ~gOti3' which implies 9 (t) ~ e~(t-tl)g (tI) for t ~ ti' Hence (2.115) implies

112

2.

KAHLER-RICCI FLOW

by taking 'Y (t) to be a minimal geodesic, with respect to 9 (tt), joining to X2 with speed Ii' (t)l g(t1) = ce-f;(t-tt}, where

Xl

r(

t»)-l dg(h) (XI,X2). c=r;, 1-e- nret 2-1

which is the estimate we would have obtained from (2.115) by using e;;;(t-h) ~ e;;; (t2 -t1) and taking 'Y (t) to be a minimal geodesic joining Xl to X2 with speed I'Y· (t)1 g(h) -- d9 (t1)(X1,X2) t2-t1 .

9.2. Application of the trace estimate. A beautiful application of the trace differential Harnack estimate and Perelman's no local collapsing theorem is the following uniform bound for the curvatures of a solution of the normalized Kahler-Ricci flow on a closed manifold with nonnegative bisectional curvature. This proof is due to Cao, Chen, and Zhu [49] and gives a simple proof of an estimate of Chen and Tian [87], [88], who proved convergence of the Ricci flow for the normalized Kahler-Ricci flow on closed manifolds with positive bisectional curvature (Theorem 2.70). THEOREM 2.92 (NKRF: Kc (V, W) ;:::: 0 curvature estimate). If(Mn,go) is a closed Kiihler manifold with !f:: [wo] = 27rCl (M) and nonnegative bisectional curvature, then the solution g (t) to the normalized K iihler-Ricci flow, with g (0) = go, has uniformly bounded curvature for all time. PROOF. Without loss of generality we may assume the solution is nonflat and the average scalar curvature r is equal to n, independent of time. Given any time t > 1, there exists a point y E M such that R (y, t + 1) = n. By (2.116) we have for any X E M

n R(x,t)

=

R(y,t+1) 1-e- t (d;(t)(X,y)) R(x,t) ;:::: 1- e-(t+1) exp -4(1- e- l )

Since 11~e(t:1) ~ ~=:=~ = 1 + e- l for t ;:::: 1, we conclude for any (2.117)

.

X

E M,

9. MATRIX DIFFERENTIAL HARNACK ESTIMATE

113

In particular, if x E B t (y, 1) , then

R (x, t) :S n (1

+ e- 1) exp

(4 (1 ~

e- 1))

.

Hence, since g (t) has nonnegative bisectional curvature, the curvature of g (t) is bounded9 in Bg(t) (y, 1). By Perelman's no local collapsing theorem, which holds for a solution to the normalized Kahler-Ricci flow since its statement is scale-invariant and since the corresponding solution to the Kahler-Ricci flow must blow up in finite time, we conclude that there exists a constant K, > 0 depending only on the initial metric such that Volg(t) (Bg(t) (y, 1)) 2:

K,.

In particular, K, > 0 is independent of t > 1 and the choice of y E M such that R (y, t + 1) = n. We may obtain a uniform diameter bound for g (t) by Yau's argument. In particular, let x E M be a point with d (x, y) = d 2: 2. Since Rc 2: 0, by the Bishop-Gromov relative volume comparison theorem, we have (2.118) Volg(t) (Bg(t) (x, d + 1)) - Volg(t) (Bg(t) (x, d - 1)) (d + It - (d - It Volg(t) (Bg(t)(x,d-l)) :S (d-lt

C(n)

< - -d - . Since Bg(t) (y, 1) c Bg(t) (x, d + 1) \Bg(t) (x, d - 1) and Bg(t) (x, d - 1) C Bg(t) (y, 2d - 1) , by (2.118), we have Volg(t) (M) 2: Volg(t) (Bg(t) (y, 2d - 1))

2: Volg(t) (Bg(t) (x,d -1)) 2:

Volg(t) (Bg(t) (y, 1)) C (n) d.

Taking d = diamg(t) (M) , we have

. C (n) Volg(t) (M) C (n) Volg(t) (M) dlamg(t) (M) < --~--=::...:~--:- < . - Volg(t) (Bg(t) (y, 1)) K, Since under the normalized flow, Volg(t) (M) is constant, we obtain a uniform upper bound C for the diameter of g (t). Hence (2.117) implies

R (x, t) :S n (1 + e-1 ) exp ( 4 (1 ~2e- 1 )) which is our desired uniform estimate for R.

,

o

9We actually only need an upper bound on the scalar curvature for Perelman's no local collapsing theorem (Theorem 6.74).

114

2. KAHLER-RICCI FLOW

9.3. Proof of the matrix Harnack estimate. In this subsection we prove Theorem 2.87. Let

The following computation is the one corresponding to Proposition 2.79. PROPOSITION 2.93 (Evolution of the Harnack quantity Zo.(3)' Suppose that a vector field X satisfies

--

1

'\lJX'Y = '\l oXy = R'YJ + i 9'Y J, '\l oX'Y = '\lJXy = 0, and

Then Zo.13 = Z (X)o.13 defined by (2.111) satisfies the evolution equation:

(:t - Do) Zo.13 = Ro.13'YJZ,yo - ~ (Ro.1Z'Y13 + R'Y13 Zo.1) (2.119)

+ po.J'YPo131 -

Po.1JP13'Yo -

2

i Z o.13 '

The proposition follows from Lemmas 2.94 and 2.95 below. Assuming the proposition, we now prove the differential Harnack estimate, i.e., Proposition 2.93. Applying Zo.13 to a (1, O)-vector field W, we have the following general formula:

(:t - Do) (Zo.13W o. w13 ) = ( (:t - Do) Zo.(3) Wo.W13 + Zo.13 ((:t - Do) (wo.W13)) + '\lTZo.13'\lf (Wo.W13) + '\l fZo.13'\lT (Wo.W13)

,

where W13 ~ W.B. Thus, if we have a null vector W of Zo.13 at a point (xo, to) and if we extend W locally in space and time so that at (xo, to) (2.120) (2.121)

W 0:,1--'?i = Wa ,fJ = 0, (J

(~ at - Do) Wo. = 0'

9. MATRIX DIFFERENTIAL HARNACK ESTIMATE

115

where Wa = 9,6aW,6, then (2.119) implies

(:t - Ll) (ZQ~WaW,6) = (RQ~1'8Zc5'Y) Wa W,6

+ (PQ8I'P£~'Y - PQ8'YPc5~I') Wa W,6

1

2

- 2 (RQ'YZI'~ + ZQ'YRI'~) Wa W,6 - tZQ~WaW,6 = RQ~1'8Zc5'YWaW,6 + IMI'812 -IMI'c512. Here and Now we use the facts that ZQ~ ~ 0 (at least on all of M x [0, toD and W is a null vector of ZQ~. An algebraic fact, similar to Lemma 2.86 and using a second variation computation similar to that in the proof of Theorem 2.78, shows that

RQ~1'8Zc5'YWaW,6 ~ IMI'812 -IMI'c512. Thus at a point where W satisfies (2.120)-(2.121), we have

Hence, by the maximum principle, on a closed manifold we have ZQ~ ~ 0 on all of space and time. In the complete noncom pact case, one can adapt the proof in Part II of this volume of Hamilton's matrix Harnack estimate for complete solutions to the Ricci flow with nonnegative curvature operator to this Kahler setting without significant modifications. Now we give the two lemmas which are needed to complete the proof of Proposition 2.93. LEMMA

2.94.

(:t - Ll) (LlRQ~ + RQ~1'8R;yc5) 1

1

= 2LlRQpRp~ + 2RQpLlRp~ + 2RQP'I'Rp~,'Y 1 + 2Rc5'Y (\lJ\lI'RQ~

+ \l1'\l8RQ~)

1

-

Ll (RQ'YRI'~)

+ 2(\lJ\lI'RQ~+ \ll' \l8RQ~-RQPI'8Rp~-RQpRp~1'8)Rc5'Y (2.122)

+ 2RQ~1'8Rc5pRP'Y

a

+ RQ~1'8 at Rc5'Y.

2. KAHLER-RICCI FLOW

116

PROOF.

Let hOtij be a real (1, I)-tensor. Using (2.33), we compute

%t ('V s'V-yhOtij) =

(2.123)

'VS'V -y (%t hOtij) - 'V s ( (%t

r~Ot ) hpij) -

(%t

r~/3 ) 'V -yhOtp

= 'VS'V-y (%thOtij)

+ 'VS (gPu'V-yROtuhpij) + gUP'VSRiju'V-yhOtP

= 'VS'V-y (%thOtij)

+ 'VS'V-yROtphpij

+ 'V-yROtp'VShpij + 'VSRpij'V-yhOtp. Taking the complex conjugate of (2.123), we have

:t

(2.124)

('V-Y'VShOtij) = 'V-y'VS (%thOtij)

+ 'V-Y'VSRpijhOtp

+ 'VSRpij'V-yhOtp + 'V-yROtp'VShpij' Next we compute, using (2.123), (2.124) and tracing, that

a at

a [g-Ys ('Vs'V-yhOtij+'V-y'VshOtij) ] (~hOtij) = '12 at 1

=

-

a

'2 g-Yo at

('Vs'V-yhOtij

+ 'V-y'VshOtij)

1

+ '2 Ro;y ('V S'V -yhOtij + 'V -y 'V ShOtij) =

~ (:t hOtij ) + ~Ro;y ('Vs'V-yhOtij + 'V-y'VshOtij) 1

+ '2 ('V;y 'V -yROtphpij + 'V -yROtp'V ;yhpij + 'V ;yRpij'V -yhOtp) 1

+ '2 ('V-y'V;yRpijhOtp + 'V;yRpij'V-yhOtp + 'V-yROtp'V;yhpij). In particular, simplifying and taking hOtij to be the Ricci tensor, we have

(! - ~) (~ROtij) = ~ (ROtij-ySRo;y) - ~ (ROt;yR-yij) 1

+ '2 Ro;y ('V S'V -y ROtij + 'V -y 'V SROtij ) 1

+ '2 ('V;y 'V -yROtpRpij + 'V -y 'V ;yRpijROtp) + 'V -yROtp 'V ;yRpij + 'V ;yRpij 'V -yROtp, On the other hand, using (2.103), we compute

a

at (ROtij-ysR;yo)

= ('V -y 'V SROtij - ROtpRpij-yS) R;yo

+ 2ROtij-ySRopRp;y.

a

+ ROtij-yS at R;yo

9. MATRIX DIFFERENTIAL HARNACK ESTIMATE

117

Hence

(:t -

Ll ) ( LlRai3 + Rai3-y;SR-yo)

1

= 2 (\71 \7 -y R apRpi3 + \7 -y \71 Rpi3 Rap ) + 2\7 -yRap \7 ;yRpi3 1

+ 2RO;Y (\7;S\7-y R ai3 + \7-y\7;SRai3) +

(\7 -y \7 ;SRai3 - Rap Rpi3-Y;S ) R-yo

Ll (Ra;yR-Yi3)

a

+ Rai3-Y;S at R-yo

+ 2Rai3-y;SRopRP"Y. Finally we obtain (2.122) from the commutator equations: 1

2 (\7-y\7;SRai3 -

\7-y\7;SRai3) =

1

-2 (R-y;SapRpi3 -

R-y;Spi3Rap)

and \7;y \7 -yRap - \7 -y \7 ;yRap

= - (R-y-yauRup + R-y-ypuRau) = RauRup - RupRau = 0,

which imply

o LEMMA

2.95.

(:t -

Ll ) (\7 -y R ai3 X -Y)

1

2 (Rap \7-yRpi3 + \7-y RapRpi3 + \7 pRai3R-yp) X-Y + (\7 pRauR-Yi3up - Rau-yp \7 pRui3) X-Y + \7-y (Rai3puRup) X-Y

= -

(2.125)

+ \7-yRai3

PROOF.

(:t - Ll) X-Y - \7;S\7-yRai3\7oX-Y - \7o\7-yRai3\7;SX-Y.

We compute using (2.105)

(:t - Ll)

\7 -y R ai3 = \7 -y

(:t - Ll) Rai3 +

(\7 -yLl - Ll \7 -y) Rai3

- (!r~a) Rpi3 = \7-y (Rai3p;SRop - RapRpi3) + gpif\7-yRau Rpi3 (2.126)

1

+ 2 (-Rui3\7 -yRau + Rau \7-y R ui3) - R-ypau \7 pR ui3 + R-ypui3 \7 pRau -

1

2R-yu \7 uRai3'

2. KAHLER-RICCI FLOW

118

Here we also used (2.33) and the following general identity: 1 (V'-yDo. - Do. V' -y) haj3 = "2 V' p (V' -y V' p - V' pV' -y) haj3 1

+ "2 (V'-y V' p 1

V' pV' -y) V' phaj3

R-ypau h qj3

+ R-Y{iuj3 h au)

+ "2 (- R-yu V' qh a j3 -

R-ypau V' phqj3

= "2 V' p ( 1

(2.127)

+ R-ypqj3 V' phau )

1

= "2 (-V'-y R au hqj3 + V' -yRqj3hau) - R-ypau V' phqj3

1

+ R-ypqj3 V' phau - "2 R-yu V' qhaj3'

where we used the second Bianchi identity (2.9). Simplifying (2.126), we have

(%t - Do.) V'-y R aj3 = - ~ (Rqj3 V'-yRau + Rau V'-yRqj3 + R-yu V'qRaj3) + R-ypqj3 V' pRau -

R-ypau V' pRqj3

+ V' -y (Raj3pSR8p)

.

Equation (2.125) now follows from this and the general formula

(%t - Do.) (V'-y R aj3X-Y) = [ (%t - Do.) (V'-y R aj3) ] X-y + V'-y R aj3 (%t - Do.) X-y - V'SV'-y R aj3V'8 X -Y - V'8V'-y R aj3V'SX-Y.

o EXERCISE

2.96. Prove Proposition 2.93 using Lemmas 2.94 and 2.95.

10. Linear and interpolated differential Harnack estimates In this section we consider a differential Harnack estimate related to the estimate of H.-D. Cao considered in the previous section. This estimate has applications in the study of the geometry and function theory of noncom pact Kahler manifolds with nonnegative bisectional curvature. Let (M n , 9 (t)), t E [0, T), be a complete noncompact solution of the Kahler-Ricci flow with bounded nonnegative bisectional curvature. By Shi's theorem, given an initial metric which is complete with bounded nonnegative bisectional curvature, such a solution exists, at least for some short time T > 0, with

IV' Rm (x, t) I :s

c

t 1/ 2

for some C < 00. Analogous to the Riemannian case (see Remark 2.25 in this chapter or Theorem 10.46 on p. 415 of [111]), we consider a solution to the linearized Kahler-Ricci flow. That is, we let haj3 be a Hermitian symmetric (1, I)-tensor satisfying the Kahler-Lichnerowicz Laplacian heat equation:

10.

LINEAR AND INTERPOLATED DIFFERENTIAL HARNACK ESTIMATES

119

(2.128)

A bound for reasonable solutions is given by the following result (see Lemma 1.2 and Proposition 1.1 in Ni and Tam [290]). PROPOSITION 2.97 (Exponential bound for h Ot i3)' Suppose a solution hOti3 of (2.128) satisfies, for some constants A and B, the following inequalities:

IhOti3 (x, 0) I ::; eA (1+ r o(x))

(2.129) and (2.130)

Then there exists a constant C <

00

such that

IhOti3 (x, t) I ::; eC (1+r o(x)). Furthermore, if (hOti3 (x, 0)) 20, then (hOti3 (x, t)) 2 0 for all t > O. The analogue of the linear trace differential Harnack estimate for the Riemannian Ricci flow, Theorem A.57, is as follows (see Theorem 1.2 on p. 633 of Ni and Tam [290]). The Kahler linear trace differential Harnack quadratic is defined by (compare with (A.27))

Z (h, V)

~ ~gOti3 (V'13 div (h)Ot + V'

01

div (h)i3)

+ ROti3h/3a.

+ gOti3 ( div (h)Ot Vi3 + div (h)i3 VOl) + hOti3 V/3 Va. + ~, where V is a vector field of type (1,0), H ~ gOti3 hOti3' and (2.131)

div (h)Ot ~ g'Yi3V''YhOti3'

THEOREM 2.98 (Kahler linear trace differential Harnack estimate). Suppose that (Mn,g(t)), t E [O,T), is a complete solution of the Kahler-Ricci flow with bounded nonnegative bisectional curvature and (hOti3) 20 is a solution of the Kahler-Lichnerowicz Laplacian heat equation (2.128) satisfying (2.129) and (2.130). Then Z (h, V) 20 on M x [0, T) for any vector field V of type (1,0) . The proof of this theorem requires a number of calculations which we state and prove. In these calculations the theme is to derive a heat-type equation for each of the quantities under consideration. We then need to combine terms in a good way so that we obtain a supersolution to the heat equation. The way this is accomplished, as in the Riemannian case, is to look for terms which vanish on gradient Kahler-Ricci solitons.

2. KAHLER-RICCI FLOW

120 LEMMA

2.99. We have the following equations and their complex conju-

gates: (1) (2.132)

(:t - ~)

div (h)a

(:t - ~)

(ga J3 \7J3 div (h)a)

= R/LiJ \7//ha{.L + h{.L// \7 aR/LiJ -

~RaiJ div (h)//,

(2) (2.133)

= R/La \7 {.L div (h)a + \7 aR/LiJ \7//ha{.L

+ \7 aR/LiJ \7 ah{.L// + R/LJ\7 a \7//ha{.L + h{.L// \7 a \7 aR/LiJ. PROOF.

Using (2.128) and (2.131), we compute

(! - ~) \7-y haJ3 = \7 (:t - ~)haJ3 + (\7-y~ -y

= \7 -y ( RaJ3e8hU 1

-

~ \7-y) haJ3 -

(:t r~a )

hliJ3

~ (RaeheJ3 + ReJ3hae) ) 1

- "2 \7 -y R a8 hliJ3 + "2 \7 -yRliJ3 h a8 - R-Yfia8\7.,.,hliJ3

+ R-yfiliJ3\7.,.,ha8 -

1 "2R-Y8\7lihaJ3

+ \7 -yRa8hliJ3'

since

2 (\7 -y~

-

~ \7 -y) haJ3

= \7-y (\7.,., \7 fi + \7 fi \7.,.,) haJ3 - (\7.,.,\7 fi + \7 fi \7.,.,) \7 -yhaJ3 = \7.,., (\7 -y \7 fi - \7 fi \7 -y) haJ3 + (\7 -y \7 fi - \7 fi \7 -y) \7.,.,haJ3

+ R-YfiliJ3 ha8 ) R-Yfia8\7.,.,hliJ3 + R-YfiliJ3\7.,.,ha8

= \7.,., ( - R-yfia8 hliJ3

- R-Yfi.,.,8\7li h aJ3 = -\7-yRa8hliJ3 + \7 -yRliJ3ha8 - 2R-Yfia8\7.,.,h liJ3 + 2R-YfiliJ3\7.,.,ha8 - R-y8\7lihaJ3· Simplifying, we have

(:t - ~)

\7-y h aJ3

= \7-y R aJ3e8 hlie - R-Yfia8\7.,.,hliJ3 + R-YfiliJ3\7.,.,ha8 1

+ RaJ3e8\7 -yhlie - "2 (Rae\7 -y heJ3 + ReJ3\7-yhae + R-Y8\7lihaJ3) . Since div (h) a

= g-YJ3\7-yhaJ3' taking the trace and cancelling terms, we have

(:t - ~)

div(h)a = g-yJ3

(:t - ~)

\7-y h aJ3 + R/3'Y\7-y h aJ3

= \7 a Re8 h lie + Rlifi \7.,.,ha8 -

which is (2.132).

1

"2 Rae\7 {3heJ3'

10. LINEAR AND INTERPOLATED DIFFERENTIAL HARNACK ESTIMATES

121

Next we verify (2.133). Using (2.132) and (2.21), we compute

(:t - ~) = (:t - ~)

(V 13 div (h)a)

V 13

div (h)a

= V 13 ( Rf..liJ V vhajl

+ (V t3~ -

+ hjlv VaRf..liJ -

~ V13) div (h)a

~RaiJ div (h)v)

+ ~V t3RJa div (h)8 - ~Rt38VJ div (h) a + Rt3,Ja Vi div (h)8 = V t3Rf..liJ V vhajl + V t3hjlV VaRf..liJ + Rf..liJ V 13 V vha'i + hjlV V 13 V aRf..liJ -

~ RaiJ V13 div (h) v

~Rt38 VJ div (h)a + Rt3,Ja Vi div (h)8·

Tracing and cancelling terms, we have

(:t - ~) (:t - ~)

(gat3V 13 div (h)a)

= gat3

(V 13 div (h)a)

+ R,Ba V 13 div (h) a

= V aRf..liJ V vhajl + V ahjlV V aRf..liJ + R,Ba V 13 div (h) a + Rf..liJ V a V vhail

+ hilv Va VaRf..liJ, D

which is (2.133).

EXERCISE 2.100. Write down the formulas for the complex conjugate equations to (2.132) and (2.133).

Now let for c

>0

and

hat3 ~ hat3 + cgat3· Since Z is of the form Z = A+BaVa +BaVa + hat3 Va V,B and hat3 2: cgat3 > 0, at each (x, t) we have that Z attains its minimum for some V. By taking the first variation, we immediately see that (2.134) Differentiating this, we have (2.135) (2.136)

+ V ,Bhai V, + hai V,B V, = 0, V 13 div (h)a + V t3hai V, + hai V 13 V, = o.

V,B div (h)a

122

2. KAHLER-RICCI FLOW

In each of the above instances, we also have the complex conjugate equations; we leave it to the reader as an exercise to write these down. Recall that Cao's Kahler matrix differential Harnack quadratic is (2.111): Za~

=

fl.Ra~

Ra~

+ Ra~-y8Ry6 + V-yRa~V-y + V-yRa~V-y + Ra~-y8V-yV6 + -t-·

From (2.132) and (2.133) and their conjugate equations, while substituting in (2.134), (2.135), (2.136) and their conjugate equations, we obtain

(~ at -fl.) Z = Z aJJf-Ih-a aJJ + h -yu:«v (2.137)

1( - t -h-y8 (V6 V-y

a

V--y - R a-y- - ~g t a-y-)

(v-v" a

u

R-r au - ~g-r\ t au)

+ V-yV6) + 2R6-yh-Y8 + 2 (H t+ en) + 2eR)

+ h-Y8 V Q V-y Va V6 .

From (2.134) and the trace of (2.136), we have ~ 11Z = Ra~hQ/3 - 2ha~VQV/3 - 2h/3QVaV~

Substituting this into the

RHS

+

H

+ en t

+eR.

of (2.137) yields

Z aJJf-Ih-a+h --~g Oi.JJ -yu:«v a V--R -y a-y t a-y-)

(V-V"-R-r-~g-r) a u au t au

>0 - .

Hence we have for the minimizer V satisfying (2.134)

(:t - fl.) (t z) 2: 2

O.

By applying the maximum principle (see pp. 639-640 of [290] for details), we may conclude that t 2 Z 2: 0 for all t > O. We have the following matrix differential Harnack estimate due to one of the authors [287]. THEOREM 2.101 (Matrix interpolated differential Harnack estimate). Let (Mn, 9 (t)) be a complete solution of the e-speed Kahler-Ricci flow

(2.138) where e > 0, with bounded nonnegative bisectional curvature, and let u be a positive solution of the forward conjugate heat equation au (2.139) at = fl.u + eHu. Then for any (1, O)-form V we have

(2.140) Equivalently, f (2.141)

~

log u satisfies fa~

1

+ eRa~ + tga~ 2: O.

10. LINEAR AND INTERPOLATED DIFFERENTIAL HARNACK ESTIMATES

123

PROOF. First we observe that the equivalence of (2.140) and (2.141) follows from the fact that the minimizing {I, O)-form Va for the LHS of (2.140) is equal to -~ and dividing (2.140) by u. We compute

af 2 at = b.f + IY' -til + ER. Using (2.22) and commuting a pair of derivatives, we have

+ IY'_rf12 + ER) = b.LfOt.i3 + EY' a Y'i3 R + f a,rfi3;y + f a;yfi3"Y + Y';yfY'"Yfai3 + Y'"YfY' ;yfOt.i3 + Ra;yoi3Y'JfY'"Yf.

:t fai3 = Y' a Y'i3 (b.f (2.142)

Using the analogue of (2.105) for the E-speed Kahler-Ricci flow,

a

at ROt.i3 = Eb.LR ai3 = EY' a Y' i3 R , we then compute

:t

(fOt.i3

+ EROt.i3 +

t

+ EROt.i3) + fa"Yfi3;Y + fOt.;yfi3"Y + Y';yfY'''Yfai3 + Y'''YfY';yfai3 + Ra;yoi3Y'JfY'''Yf + E2 b.ROt.i3 + E2 ROt.i3"YJRo;y - E2 Ra;yR"Yi3

9ai3) = b.L (JOt.i3

1 E - t29Ot.i3 - Rai3'

t

Hence letting SOt.{3 ~ fai3

+ ERai3 + t9ai3

(! - b.L) SOt.i3

+ E2

=

fa"Yfi3;Y

and using (2.106), we have

( b.Rai3 + Rai3"YJ R O;Y +

:t

Rai3 )

- EY';yfY'''YROt.i3 - EY'''YfY';yROt.i3 + Ra;yoi3Y'JfY'''Yf + Y';yfY'"Y S ai3 + Y'"YfY' ;ySOt.i3

+ ~Sa;y (fi3"Y +~

ER"Yi3 - t9"Yi3 )

(fa;y - ERa;y - t9a;y) Si3"Y'

Now for the E-speed Kahler-Ricci flow, by Cao's Kahler matrix differential Harnack estimate (2.111) with Xa = -~Y'af, we have

1

o ::; b.Rai3 + Rai3"YJ R o;y + Et ROt.i3 1 1 - -Y';yfY'''YROt.i3 - -Y'''YfY';yRai3 E

E

1 + 7. Ra;yoi3 Y' J fY' "Y f. E

124

2.

KAHLER-RICCI FLOW

Hence

(:t -

b.. L ) 8 a(3 2 fa,d(3'Y +

+ \!'Yf\!"(8a(3 + \!"(f\!'Y8a(3

~8a'Y (f(3"( -

+~

cR"((3 - t g"((3)

(fa'Y - cRa'Y - tga'Y) 8(3"(.

The estimate (2.141) follows from an application of the maximum principle; 0 see [287] for details. Tracing (2.140), i.e., multiplying by ga(3 and summing, we have COROLLARY 2.102 (Trace interpolated differential Harnack estimate). Under the hypotheses of Theorem 2.101, nu b..u + cuR + + gaf3 (uaV(3 + u(3Va + uVaV(3) 20,

t

which, by taking Va = - ~ and then dividing the resulting expression by u, implies the equivalent inequality n b..log u + cR + -t > - O.

J

Let N ~ M u log udf..t be the (classical) entropy of u. We have under (2.139), $tdf..t = -cRdf..t (since $t detg"(8 = -cRdetg"(8)' and

~ = 1M (b..u + cRu + (logu) b..u) df..t

1M (b..logu+cR)udf..t 2 -!f 1M udf..t.

=

In other words, (2.143) 11. Notes and commentary Some books containing material on or devoted to complex manifolds and Kahler geometry, in essentially chronological order, are Weil [370], Chern [95], Goldberg [157], Kobayashi and Nomizu [236], Morrow and Kodaira [275], Griffiths and Harris [166]' Aubin [13]' Kodaira [238]' Besse [27], Siu [334]' Mok [270]' Tian [347]' Wells [371], and Zheng [383]. We refer the reader to these books for the proper study of Kahler geometry. For the Ricci flow on real2-dimensional orbifolds, see L.-F. Wu [372] and [112]. For the Ricci flow on noncompact Riemannian surfaces, see Wu [373]'

11. NOTES AND COMMENTARY

125

Daskalopoulos and del Pino [119]' Hsu [206], [207], and Daskalopoulos and Hamilton [120].

CHAPTER 3

The Compactness Theorem for Ricci Flow Although this may seem a paradox, all exact science is dominated by the idea of approximation. - Bertrand Russell

The compactness of solutions to geometric and analytic equations, when it is true, is fundamental in the study of geometric analysis. In this chapter we state and prove Hamilton's compactness theorem for solutions of the Ricci flow assuming Cheeger and Gromov's compactness theorem for Riemannian manifolds with bounded geometry (proved in Chapter 4). In Section 3 of this chapter we also give various versions of the compactness theorem for solutions of the Ricci flow. Throughout this chapter, quantities depending on the metric gk (or gk (t)) will have a subscript k; for instance, 'V'k and Rmk denote the Riemannian connection and Riemannian curvature tensor of gk. Quantities without a subscript depend on the background metric g. Often we suppress the t dependence in our notation where it is understood that the metrics depend on time while being defined on a space-time set. Given a sequence of quantities indexed by {k}, when we talk about a subsequence, most of the time we shall still use the indices {k} although we should use the indices

{jk} .

1. Introduction and statements of the compactness theorems

Given a sequence of solutions (Mr, gk (t)) to the Ricci flow, Hamilton's Cheeger-Gromov-type compactness theorem states that in the presence of injectivity radii and curvature bounds we can take a Coo limit of a subsequence. The role of the compactness theorem in Ricci flow is primarily to understand singularity formation. This is most effective when the compactness theorem is combined with monotonicity formulas and other geometric and analytic techniques, in part because these formulas and techniques enable us to gain more information about the limit and sometimes enable us to classify singularity models. This has been particularly successful in low dimensions. In latter parts of this volume we shall see some examples of 127

3. THE COMPACTNESS THEOREM FOR RICCI FLOW

128

this:

ICompactness Theorem I INo local collapsing I 1

+---

I]\10notonicity I

./

ISingularity analysis I In general, there are three scenarios in which we shall apply the compactness theorem for the Ricci flow. The compactness result may be applied to study solutions (M n , 9 (t)) to the Ricci flow defined on time intervals (0', w) , where w ::; 00 is maximal, i.e., the singularity time. To understand the limiting behavior of the solution 9 (t) as t approaches w, we shall take a sequence of times tk ~ wand consider dilations of the solution 9 (t) about the times tk and a sequence of points Ok E M by defining (3.1)

gk (t) = Kkg (tk

+ KJ:1 t ) ,

where Kk = IRm (Ok, tk)1 is the norm of Rm (g (tk)) at the point Ok. We are interested in determining when there exists a subsequence of pointed solutions to the Ricci flow (M, gk (t) , Ok) which limits to a complete solution (M~, goo (t) ,000 ) . This limit solution reflects some aspects of what the singularity looks like near (Ok, tk). Similarly, when 0' = -00 for solution (M, 9 (t)), which arises when we already have a (first) limit solution of a finite time singularity, we may consider sequences tk ~ -00 and take a second limit, now backward in time. Yet other limits that we shall consider arise from dimension reduction on a limit solution. Here tk remains fixed whereas Ok tends to spatial infinity. Many of the topics in this volume are related to the study of the geometry (and topology) of the limits of these solutions when they exist. 1.1. Definition of convergence. Now we review the definition of Coo_ convergence on compact sets in a smooth manifold Mn. By convergence on a compact set in CP we mean the following.

3.1 (CP-convergence). Let K c M be a compact set and let {gdkEN' goo, and 9 be Riemannian metrics on M. For p E {O} UN we say that gk converges in CP to goo uniformly on K if for every c > 0 there exists ko = ko (c) such that for k ~ ko, DEFINITION

sup sup IV o (gk - goo)lg

< C,

O::;o::;pxEK

where the covariant derivative V is with respect to g. Note that since we are on a compact set, the choice of metric 9 on K does not affect the convergence. For instance, we may choose 9 = goo. In regards to Coo-convergence on manifolds, with the noncompact case in mind, we have the following. We say that a sequence of open sets {UdkEN in a manifold Mn is an exhaustion of M by open sets if for any compact set K c M there exists ko EN such that Uk ::) K for all k ~ ko.

1.

INTRODUCTION; STATEMENTS OF THE COMPACTNESS THEOREMS

129

DEFINITION 3.2 (CC)()-convergence uniformly on compact sets). Suppose {UdkEN is an exhaustion of a smooth manifold Mn by open sets and gk are Riemannian metrics on Uk. We say that (Uk,gk) converges in Coo to (M, goo) uniformly on compact sets in M iffor any compact set K c M and any p > 0 there exists ko = ko (K,p) such that {gdk>k _ 0 converges in CP to goo uniformly on K. In order to look at convergence of manifolds which come from dilations about a singularity, we must ensure that the form of convergence can handle diameters going to infinity. When this happens, a basepoint, or origin, is carried along with the manifold and the Riemannian metric to distinguish what parts of the manifolds in the sequence we are keeping in focus. This allows us to compare spaces that either have diameters going to infinity or are noncompact. DEFINITION 3.3 (Pointed manifolds and solutions). A pointed Riemannian manifold is a 3-tuple (Mn, g, 0), where (M, g) is a Riemannian manifold and 0 E M is a choice of point (called the origin, or basepoint). If the metric g is complete, the 3-tuple is called a complete pointed Riemannian manifold. We say that (Mn, g (t), 0), t E (a,w) , is a pointed solution to the Ricci flow if (M,g(t)) is a solution to the Ricci flow. REMARK 3.4. In [187] Hamilton considered marked Riemannian manifolds (and marked solutions to the Ricci flow), where one is also given a frame F = {e a } at 0 orthonormal with respect to the metric g (0) with o E (a,w). Since for most applications, the choice of frame is not essential, we restrict ourselves to considering pointed Riemannian manifolds in this chapter.

:=1

Convergence of pointed Riemannian manifolds is defined in a way which takes into account the action of basepoint-preserving diffeomorphisms on the space of metrics. DEFINITION 3.5 (COO-convergence of manifolds after diffeomorphisms).

A sequence {(Mk,gk,Ok)}kEN of complete pointed Riemannian manifolds converges to a complete pointed Riemannian manifold (M~, goo, 0 00 ) if there exist

(1) an exhaustion {UkhEN of Moo by open sets with 0 00 E Uk and (2) a sequence of diffeomorphisms q,k : Uk ---t Vk ~ q,k (Uk) C Mk with q,k (0 00 ) = Ok such that (Uk,q,t: [gklvk]) converges in Coo to (Moo,goo) uniformly on compact sets in Moo. We shall also call the above convergence Cheeger-Gromov convergence in Coo. The corresponding definition for sequences of pointed solutions of the Ricci flow is given by the following.

130

3.

THE COMPACTNESS THEOREM FOR RICCI FLOW

DEFINITION 3.6 (COO -convergence of solutions after diffeomorphisms).

A sequence {(Mk' 9k (t) ,0k)}kEN' t E (a, w) , of complete pointed solutions to the Ricci flow converges to a complete pointed solution to the Ricci flow (M~, 900 (t) ,000 ) , t E (a, w) , if there exist (1) an exhaustion {UdkEN of Moo by open sets with 0 00 E Uk, and (2) a sequence of diffeomorphisms
(1) (uniformly bounded geometry) IV~Rmklk:::; Cp for all p 2:: 0 and k where C p < independent of k and

on Mk 00

is a sequence of constants

1.

INTRODUCTION; STATEMENTS OF THE COMPACTNESS THEOREMS

131

(2) (injectivity radius estimate) injgk (Ok) 2: for some constant

/'0

/'0

> O.

Then there exists a subsequence {jd kEN such that {(Mjk,9jk,Ojk)}kEN converges to a complete pointed Riemannian manifold (M~, 900,000 ) as k~ 00.

For sequences of solutions to the Ricci flow the corresponding convergence theorem takes the following form. THEOREM 3.10 (Compactness for solutions). Let {(Mk' gk (t), Ok)}kEN' (a, w) 3 0, be a sequence of complete pointed solutions to the Ricci flow such that

t

E

(1) (uniformly bounded curvatures) IRmklk ~ Co

on Mk x (a,w)

for some constant Co < 00 independent of k and (2) (injectivity radius estimate at t = 0) injgdO) (Ok) 2: for some constant

/'0

/'0

> O.

Then there exists a subsequence {jkhEN such that {(Mjk' 9jk (t) ,0jk)}kEN converges to a complete pointed solution to the Ricci flow (M~, goo (t) ,000 ), tE (a,w), as k~oo. . Note that the second theorem only supposes bounds on the curvature, not bounds on the derivatives of the curvature. This is because, for the Ricci flow, if the curvature is bounded on (a, w) , then all derivatives of the curvature are bounded at times t > a (see Chapter 7 of Volume One or Theorems A.29 and A.30 of this volume).l In particular all derivatives of the curvature are bounded at time t = 0 and we can apply Theorem 3.9 to {(M k, gk (0), Ok) hEN· In the next section we follow the proofs of Hamilton in [187]. We shall assume Theorem 3.9, which will be proven in Chapter 4. We will show that if there is a subsequence such that (M k,9k (0) ,Ok) converges to a complete limit (M~, 900 (0) ,000 ) , then there is a subsequence (Mk' gk (t) ,Ok) which converges at all times. 1The bounds on the derivatives of Rm get worse as t

-+

Q.

3. THE COMPACTNESS THEOREM FOR RICCI FLOW

132

2. Convergence at all times from convergence at one time In this section we give the proof that the compactness theorem for Ricci flow (Theorem 3.10) follows from the compactness theorem at time t = (Theorem 3.9). This is done by showing that bounds on the metric and covariant/time-derivatives of the metric at time t = extend to bounds on the metric and covariant derivatives of the metric at subsequent times in the presence of bounds on the curvature and covariant derivatives of curvature (for all time). This is shown in subsection 2.1 below. The Arzela-Ascoli theorem is then used to show that these bounds on the covariant/timederivatives of the metric imply that a subsequence converges to a solution of the Ricci flow for all times (in subsection 2.2.2 below).

°

°

2.1. Uniform derivative of metric bounds for all time. In order to extend the convergence at one time to convergence at all times, the following derivative bounds need to be shown. LEMMA 3.11 (Derivative of metric bounds at one time to all times). Let M n be a Riemannian manifold with a background metric g, let K be a compact subset of M, and let gk be a collection of solutions to the Ricci flow defined on neighborhoods of K x [,8, 'l/J], where to E [,8, 'l/J]. Suppose that

(1) the metrics gk (to) are all uniformly equivalent to 9 on K, i. e., for all V E TxM, k, and x E K, C- 1 g (V, V) :::; gk (to) (V, V) :::; Cg (V, V) , where C < 00 is a constant independent of V, k, and x; and (2) the covariant derivatives of the metrics gk (to) with respect to the metric 9 are all uniformly bounded on K, so that

for all k and p :2: 1, where C p < 00 is a sequence of constants independent of k; and (3) the covariant derivatives of the curvature tensors Rmk (t) of the metrics gk (t) are uniformly bounded with respect to the metric gk (t) on K x [,8, 'l/J] :

(3.2)

IV'~ Rmklk :::; C~ for all k and p :2: 0, where C~ is a sequence of constants independent of k.

Then the metrics gk (t) are uniformly equivalent to 9 on K x [,8, 'l/J], e.g., (3.3)

B (t, to)-l g(V, V) :::; gk (t) (V, V) :::; B (t, to) g(V, V),

where

B (t , t a) = C e 2v'n-1Cblt-tol ,

2.

CONVERGENCE AT ALL TIMES FROM CONVERGENCE AT ONE TIME

133

and the time-derivatives and covariant derivatives of the metrics gk (t) with respect to the metric 9 are uniformly bounded on K x [.8, '1/1], i.e., for each (p, q) there is a constant Cp,q independent of k such that

(3.4) 1

8q gd) -'\lP t -< Cp,q 8tq

1-

for all k. REMARK 3.12. Since we often assume bounds on Rm whereas the metric evolves by Rc, we note -In=lIRmlg:::; Rc:::; In=lIRmlg· Since we often interchange 9 and gk norms, we recall the following elementary fact. LEMMA 3.13 (Norms of tensors with respect to equivalent metrics). Suppose that the metrics 9 and h are equivalent:

C-Ig:::; h:::; Cg. Then for any (p, q)-tensor T, we have

(3.5)

ITlh :::; C(p+q)/2I T lg·

PROOF. We can diagonalize 9 and h so that gij = The assumption implies C- I :::; Ai :::; C for all i. Then

bij

and

h ij

=

Aibij.

... h kpl p hidl ... hiqjqT.kl··:kPT~l···~p ITI2h = "h L...J klll tl···tq Jl···Jq

< "

L...J

Tkl··:kPT.kl··:kP. 1l···1q

1l···1q

kl···kpjil···iq

o PROOF OF LEMMA 3.11. For the first part, since

8

at gk (t) (V, V) = -2 Rc k (t) (V, V) and IRck (t) (V, V)I :::; In=lCb9k (t) (V, V), we can estimate the time-derivatives 1

8 I at loggk (t) (V, V) =

We have proved

(3.6)

1-2RC k (t)(V,V)1 ~, gk (t) (V, V) :::; 2vn - 1Co·

134

3. THE COMPACTNESS THEOREM FOR RICCI FLOW

where C ~ 2Jn=lCb. Now we compute

C It 1

-

tol

~

1:

1 1

%t log 9k (t) (V, V) 1 dt

~ 11:1 %t log9k (t) (V, V) dtl -iiog 9k (tl) (V, V) 1 9dto) (V, V) , or equivalently, e-Clt1-toI9k (to) (V, V) ~ 9k (tl) (V, V) ~ eClh-tol9k (to) (V, V). Hence we have C-le-Clt1-toI9 (V, V) ~ 9k (it) (V, V) ~ CeClt1-tol9 (V, V) . This completes the proof of (3.3). For the second part we need to estimate the space- and time-derivatives of 9k (t). We begin with estimating the first-order covariant derivatives of 9k (t) . Note that

8 V' a (9k)bc = 8x a (9k)bc -

d

d

r ab (9k)dc - r ac (9k)bd ,

so if we take the right combination, we see that

(9k)ec (V' a (9k)bc

+ V'b (9k)ac -

V' c (9k)ab)

= 2 (rk)~b - r~b - (9k)ec r~c (9khd - r~b - (9k)ec ric (9k)ad (3.7)

= 2 (rk)~b -

+ (9k)ec r~ (9k)ad + (9k)ec r~c (9k)bd

2r~b'

This implies that

(3.8) From we have

(3.9) Hence the tensors V'9k (t) and rk (t) - r are equivalent. We recall that the derivative of the Christoffel symbols (see, for instance, (6.1) on p. 175 of Volume One) is

:t (rk)~b

= -

(9k)cd [(V'k)a (RCk)bd

+ (V'k)b (RCk)ad -

(V'k)d (Rck)abl·

2. CONVERGENCE AT ALL TIMES FROM CONVERGENCE AT ONE TIME

135

So as tensors, we find that I:t (fk - r)lk ::; 3lVd Rc k)lk ::; Thus

3Jn=1C~ It

1 -

tol

~

1:

1

I:t (fk (t) - r)lk dt

~ 11: ~ Ifk

3Jn=1C~.

1

:t (fk (t) - f) dtl k

(t1) - flk - Ifk (to) - flk .

Hence we have a bound Ifk (t) - flk ::; 3Jn=1C~ It - tol (3.10)

::;

3Jn=1C~ It -

tol

+ Ifk (to) -

flk

+ ~C3/2Cl

using (3.8) and (3.5). Since It - tol ::; '!f; - f3, we have in (3.3): B (t, to) ::; B ('!f;, f3) for all t E [f3, '!f;]. Thus by (3.9) and (3.10), (3.11)

where

C1 ,0 ~ B 3 / 2 ('!f;, f3) (6Jn=lC~ ('!f; - f3) + 3C3 / 2 C 1 ) This proves (3.4) for p = 1 and q = O. Next we prove inductively that for p IVP Rc kl ::; C; IVPgkl

~

+ C;'

.

1, IVPgkl::; Cp,o

and

(where C~, C~', and Cp,o are independent of k). If p and (3.10),

=

1, then using (3.8)

+ Vk RCklk fklk IRcklk + IVk RCklk)

IVRckl (t) ::; B (t, to)3/21(V - Vk) RCk

::; B (t, to)3/2 (If ::; B (t, to)3/2 (

(3Jn=1C~ I'!f; -

If the estimates hold for p < N with N p = N. First we have

~

f31

+ ~C3/2C1) Cb + C~) .

2, then we will prove them for

N

IV N

RCkl

=

LV

N- i

(V - Vk) V1- 1 RCk

N- i

(V - Vk) V1- 1 RCkl

+ Vt' RCk

i=l

N

: ; L IV i=1

+ IVt' RCkl·

3. THE COMPACTNESS THEOREM FOR RICCI FLOW

136

Note that, using (3.7), we can rewrite V' - V'k = r - rk as a sum of terms of the form V'9k. When i = 1, we can bound IV'N-l (V' - V'k)Rckl by a sum of terms of the form IV'N- j 9kllV'j RCkl, 0 ~ j ~ N - 1. When 2 ~ i ~ N, we can bound IV'N-i (V' - V'k) V'1- 1 RCkl by a sum of terms of the form IV'N-i-j+19kllV'jV'~-1 RCkl, 0 ~ j ~ N - i. We can also bound lV'jV'1- 1 RCkl = I((V' - V'k)

+ V'k)j V'1- 1 RCkl

by a sum of terms which are products of lV'i+i-l Rc kl ' 0

~ f ~

j, and

severallV't'9kl, 1 ~ f ~ j. By the assumption of Lemma 3.11, the induction assumption and the equivalence of 1·1 and 1·l k , we get IV'N Rc k 1 ~

C'/v IV'N 9k 1+ C'jJ.

Now we turn to bounding IV'N 9kl. Since 9 does not depend on t,

8

N

N

8t V' 9k = -2V' RCk and 8 1V' N9k 12 8t

8 8 V' N 9k, V' N = 2 \/ 8t 9k) ~ 18t V' N9k 12

= 41V'N RCkl2 + IV'N 9kl 2 ~

+ 1V' N9k 12

(1 + 8 (C'/v )2) IV'N 9kl 2 + 8 (C'jJ) 2 .

Integrating the above differential inequality of 1V' N 9k 12 , we get (compare with (7.47)) IV'N 9kl 2 (t)

~ e(1+8(C~y)(t-to) (IV'N 9kl 2 (to) +

8 (C'jJ)2

).

1 + 8 (C'/v) 2

This implies IV'N 9k (t)1 ~ CN,O, and the induction proof is complete, as well as (3.4) for the q = 0 case. Note that the above proof of bounding 1V' N Rc k 1 can be used to show that IV'PV'%Rckl, IV'PV'%Rkl, and IV'PV'%Rmkl are bounded independent of k. aq aq - 1 When q ~ 1, then atq V' P9dt) = V'P atq-l (-2Rcdt)). Using the evoV'P 9k (t) is lution equation of the curvature Rm k (t), we know that bounded by a sum of terms which are products of

l%t:

1V'PI V'%l Rm k 1(t) ,

1V'PV'% Rc k I,

and

I

1V'PV'%Rk 1.

D

2.

CONVERGENCE AT ALL TIMES FROM CONVERGENCE AT ONE TIME

137

2.2. Convergence at all times from convergence at one time. 2.2.1. The Arzela-Ascoli theorem. With uniform derivative bounds on the metrics in the sequence, the compactness theorem will follow from the Arzela-Ascoli theorem. LEMMA 3.14 (Arzela-Ascoli). Let X be a a-compact, locally compact Hausdorff space. If {fkhEN is an equicontinuous, pointwise bounded sequence of continuous functions fk : X --t lR, then there exists a subsequence which converges uniformly on compact sets to a continuous function foo : X --t R The reader is reminded that a-compact simply means that the space is a countable union of compact sets, and hence any complete Riemannian manifold satisfies the assumption. COROLLARY 3.15 (Metrics with bounded derivatives preconverge). Let (Mn, g) be a Riemannian manifold and let K c Mn be compact. Furthermore, let p be a nonnegative integer. If {gd kEN is a sequence of Riemannian metrics on K such that sup O~Q:~p+1

sup 1V'Q:gkl ::; C <

00

xEK

and if there exists 8 > 0 such that gk (V, V) 2:: 8g (V, V) for all VET M, then there exists a subsequence {gk} and a Riemannian metric goo on K such that gk converges in CP to goo as k --t 00.

PROOF (SKETCH). We need to show that {(gk)behEN form an equicontinuous family. We use the fact that in a coordinate patch

V' a (gk)be

=

oxoa (9k )be - r dab (gk)de - raed (gk)bd .

Thus if IV'gkl is bounded, then Ia~a (gk)bel is bounded for each a, b, c in each coordinate patch. Hence, by the mean value theorem, the (gk)be form an equicontinuous family in the patch and there is a subsequence which converges to (goo)be. Since K is compact, we may take a finite covering by coordinate patches and a subsequence which converges for each coordinate patch. We have thus constructed a limit metric. Note that the uniform upper and lower bounds on the metrics gk ensure that goo is positive definite. Similarly, we can use the bound on 1V'2gkl to get bounds on second derivatives of the metrics Iax~~xd (9k) be I in each coordinate patch and thus show the first derivatives are an equicontinuous family. Taking a further subsequence, we get convergence in C 1 . Higher derivatives are similar. 0

2.2.2. Proof of the compactness theorem for solutions assuming the compactness theorem for metrics. We will now use Corollary 3.15 together with Lemma 3.11 to find a subsequence which converges and complete the proof of Theorem 3.10. Recall that we have assumed Theorem 3.9 and hence there is a subsequence {(Mk, gk (0), Ok)} which converges to (M~, goo, 0 00 ) .

138

3.

THE COMPACTNESS THEOREM FOR RICCI FLOW

We shall show that there are metrics 900 (t), for t E (a,w), such that 900 (0) = 900 and {(Mk' 9k (t) ,Ok)} converges to (Moo, 900 (t) ,000 ) in Coo. Since {(Mk' 9k (0) ,Ok)} converges to (Moo, 900,000 ) , there are maps
gt

3. Extensions of Hamilton's compactness theorem In this section we give several variations of Theorem 3.10. 3.1. Local compactness theorems. From the proof of the compactness Theorem 3.9 given in the next chapter and the proof of Theorem 3.10 given above, without too much difficulty one sees that a local version of Theorem 3.10 holds. In particular, we have the following. THEOREM 3.16 (Compactness, local version). Let {(M k,9k (t), Ok)}kEN' t E (a, w) :3 0, be a sequence of complete pointed solutions to the Ricci flow.

If there exist p > 0, Co < 00, and IRmklk::; Co

/'0

in

> 0 independent of k such that Bgk(o)

(Ok,p) x (a,w)

and injgdO) (Ok) ~ /'0,

then there exists a subsequence such that {(Bgk(o) (Ok,P),9k(t),Ok)}kEN converges as k ---t 00 to a pointed solution (B~,900 (t) ,000 ), t E (a,w), in Coo on any compact subset of Boo x (a, w). Furthermore Boo is an open manifold which is complete on the closed ball Bgoo(o) (000 , r) for all r < p. EXERCISE 3.17. Prove Theorem 3.16. A simple consequence of Theorem 3.16 is (see [186]) the following corollary.

COROLLARY 3.18 (Compactness theorem yielding complete limits). Let {(M k,9k(t),Ok)}kEN' t E (a,w) :3 0, be a sequence of complete pointed solutions to the Ricci flow. Suppose for any r > 0 and c > 0 there exist constants Co (r, c) < 00 such that IRm klk ::; Co (r, c)

on

Bgk(o)

(Ok, r) x (a + c, w - c)

3.

EXTENSIONS OF HAMILTON'S COMPACTNESS THEOREM

139

for all kEN. We assume injgk(O) (Ok) ~ "0 for some "0 > O. Then there exists a subsequence {(Mk' gk (t) ,Ok)} which converges to a complete solution to the Ricci flow (M~, goo (t) ,000 ) , t E (a, w) . REMARK 3.19. Note that the limit solution (M~, goo (t), 0 00 ) may not have bounded curvature. Without the injectivity radius estimate, we may use the trick of locally pulling back the solutions by their exponential maps (since the pulled-back solutions satisfy an injectivity radius estimate). We have the following. COROLLARY 3.20 (Local compactness without injectivity radius estimate). Let {(M k,gk (t) , Ok)} kEN' t E (a, w) :3 0, be a sequence of complete solutions to the Ricci flow with IRmklk ~ Co

in Bgk(o) (Ok, p) x (a,w).

Then there exists a subsequence such that {(BTokMk

(a, c) ,(expR(O))* gd t )) ,a} kEN'

where

c~ min {p,7f/JCo} ,

converges to a pointed solution (B~,goo (t) ,000 ) , t E (a,w), on an open manifold which is complete on the closed ball Bgoo(o) (000 , r) for all r < c. REMARK 3.21. There is a similar result for geodesic tubes; see §25 of Hamilton [186].

. 3.2. Compactness for Kahler metrics and solutions. Without much difficulty, the compactness theorems apply to Kahler manifolds and solutions of the Kahler-Ricci flow (see also Cao [48] and Theorem 4.1 on pp. 16-17 of Ruan [314]). THEOREM 3.22 (Compactness for Kahler metrics). Let {(M~n, gk, Ok)} be a sequence of complete pointed Kahler manifolds of complex dimension n. Suppose 1\7~Rmklk ~ Cp on Mk for all p ~ 0 and k, where Cp < 00 is some sequence of constants independent of k, and injgk (Ok) ~ "0 for some constant "0 > O. Then there exists a subsequence {jkhEN such that {(Mjk' gjk' Ojk)}kEN converges to a complete pointed complex n-dimensional Kahler manifold (M~,goo,Ooo) as k ~ 00. See the ensuing proof for the meaning of the convergence of the complex structures Jk ~ J oo . PROOF. Since Kahler manifolds are Riemannian manifolds, we can apply the Cheeger-Gromov Compactness Theorem 3.9 to obtain a pointed limit (M~,goo,Ooo) which is a complete Riemannian manifold. So the only issue is to show that the limit is Kahler. Let Jk denote the complex structure of (Mk' gk) . We have for each kEN,

(1)

Jf = -

idTM k ,

140

3.

THE COMPACTNESS THEOREM FOR RICCI FLOW

(2) (gk 0 Jk) (X, Y) ~ gk (JkX, JkY) = gk (X, Y) for all X, YET Mk, (3) V' kJk = O. Since {(Mk' gk, Ok)} converges to (Moo, goo, 0 00 ) , there are diffeomorphisms ~k : Uk ~ Vk such that ~kgk ~ goo uniformly on compact sets. From (1) we know (~klL 0 Jk 0 (~k)*' as (1, I)-tensors, are uniformly bounded on any compact set K c Moo with respect to the metrics ~kgk' Since the ~kgk are equivalent to goo on K, we conclude that (~klL 0 Jk 0 (~kt, as (1, I)-tensors, are uniformly bounded on any compact set K c Moo with respect to the metric goo. Note that (3) is equivalent to V'
((~klL 0 Jk 0 (~k)*)

= 0 for all

p.

From the proof of Corollary 3.15, there exists a subsequence (~kl L 0 Jk 0 (~k)* converging in Coo, as a (1, I)-tensor on compact sets, to a smooth map J oo : T Moo ~ T Moo. Since

(I') 0 = ((~klL 0 Jk 0 (~k)*)2 + idTMoo ~ J~ + idTM"" , (2') 0 = (~kgk) 0 ((~klL 0 Jk 0 (~k)*) - ~kgk ~ goo 0 J oo - goo, (3') 0 = V'
= O. We conclude that 0

Applying Theorem 3.10, we obtain the following corresponding result for the Kahler-Ricci flow. THEOREM 3.23 (Compactness theorem for the Kahler-Ricci flow). Let { (M~n , gk (t) , Ok) } , t E (Q, w) 3 0, be a sequence of complete pointed solutions to the Kahler-Ricci flow of complex dimension n. Suppose

IRmklk ~ Co for some constant Co <

00

on Mk x (Q,w)

independent of k and that

(Ok) ~ /'0 for some constant /'0 > O. Then there exists a subsequence of solutions such that {(Mk,gk (t) ,Ok)} converges to a complete pointed complex ndimensional solution to the Kahler-Ricci flow (M~, goo (t) ,000 ) , t E (Q, w) , as k ~ 00. injgk(O)

PROOF. By Theorem 3.10, there exists a subsequence which converges to a Riemannian solution (Moo, goo (t) ,000 ) , t E (Q,w), to the Ricci flow. From the previous theorem, (Mk, gk (0) ,Jk) converges to (Moo, goo (0) ,Joo ) as Kahler manifolds for some complex structure J oo . Now by assumption, gk (t) remains Kahler with respect to Jk. Hence gk (t) 0 Jk = gk (t) and V'gk(t)Jk = 0 for all t E (Q,w), which implies goo(t) 0 J oo = goo(t) and V'goo(t)Joo = 0 for all t E (Q,w). That is, goo (t) remains Kahler with respect to J oo . 0

3.

EXTENSIONS OF HAMILTON'S COMPACTNESS THEOREM

141

3.3. Compactness for solutions on orbifolds. Note that Definitions 3.5 and 3.6 can be easily generalized to orbifolds. (See [343] for the definition of orbifold.) We have the following generalizations of Theorems 3.9 and 3.10 to orbifolds. The version for metrics is THEOREM 3.24 (Compactness theorem for metrics on orbifolds). Let {(Mk' gk, Ok)} be a sequence of complete pointed Riemannian orbifolds and let Ek be the singular set of Mk. Suppose that (i) IV~ Rmk Ik ~ C p on Mk for all p 2:: 0 and k, where C p < 00 are constants independent of k, and (ii) Vol gk B9k (Ok, ro) 2:: vo for all k, where ro > 0 and vo > 0 are two constants independent of k. Then either of the following hold. (1) limk--+oodgk(Ok, E k ) > O. In this case there exists a subsequence {(Mk j , gkj' Ok j )} which converges to a complete pointed Riemannian orbifold (M~,goo,Ooo) with IV~oo Rmgool goo ~ C p and Volgoo Bgoo(Ooo,ro) 2:: vo. Furthermore 0 00 is a smooth point in Moo. (2) limk--+oodgk(Ok,Ek) = o. In this case there exists a subsequence {(Mkj' gkj' Okj )} such that limj--+oo d( Okj' Ekj ) = o. If we choose O~j E Ekj with dgk(Okj'O~) = d(Okj,Ekj ), then a subsequence of {(Mkj,9kj,0~j)} converges to a complete pointed Riemannian orbifold (Moo, goo, 0 00 ) with IV~oo Rmg<XJ Ig<XJ ~ C p and Volgoo Bgoo(Ooo,ro) 2:: vo· Furthermore 0 00 is a singular point in Moo.

The version for solutions of Ricci flow is the following. THEOREM 3.25 (Compactness theorem for solutions on orbifolds). Let {(M k , gk(t), Ok)}, t E (a,w), be a sequence of complete pointed orbifold solutions of the Ricci flow. Let Ek be the singular set of Mk. Suppose that (i) 1 Rmklk ~ Co on Mk x (a,w) for all k, where Co < 00 is a constant independent of k, and (ii) Volgk(o) Bgk(o) (Ok, ro) 2:: vo for all k, where ro > 0 and vo > 0 are two constants independent of k. Then either of the following hold. (1) limk--+oodgk(O) (Ok, E k ) > o. In this case there exists a subsequence {(Mkj' gk j (t), OkJ} which converges to a complete pointed orbifold solution of the Ricci flow (M~,goo(t),Ooo) with 1 Rmg""lg"" ~ Co on Moo x (a,w) and Volg<XJ(o) Bgoo(o) (000 , ro) 2:: vo. Furthermore 0 00 is a smooth point in Moo. (2) limk--+oodgk(O) (Ok, E k ) = O. In this case there exists a subsequence {(Mkj,9kj(t),Okj)} such that limj--+oo d(Okj,Ekj) = O. Furthermore if we choose O~j E Ekj with dgk(o) (Okjl O~j) = dgk(o) (Okj' EkJ, then there is a subsequence of { (Mk j , gkj (t),

O~j )}

which converges to a complete pointed

orbifold solution of the Ricci flow (M~, goo(t), 0 00 ) with 1 Rm goo Igoo ~ Co

142

3. THE COMPACTNESS THEOREM FOR RICCI FLOW

on Moo x (o:,w) and Volgoo(o) Bgoo(o) (000 , ro) singular point in Moo.

~

Vo. Furthermore 0 00 is a

Idea of the proofs. Theorem 3.25 can be proved from Theorem 3.24 in the same way as we have proved Theorem 3.10 from Theorem 3.9. On the other hand, Theorem 3.24 can be proved with some modification of the proof of Theorem 3.9 to handle the singularity (see [257]). Note that the Bishop-Gromov volume comparison theorem holds for orbifolds. Fix r > 0; for any k and qk E Mk with dgk(o) (Ok, qk) ::; r, we have Volgk(o) Bgk(o) (qk, ro) ~ VI, where VI is a positive constant independent of k but depending on Vo, r, ro, n, Co. This implies that there exists rl independent of k such that Bgk(o) (qk, rd has the orbifold topological model B n IG (qk) , where B n is the unit ball in Euclidean space centered at the origin and G (qk) cO (n) is a discrete subgroup with rank IG (qk)1 bounded independent of k. The existence of rl implies that we can modify the choice of X'lr in Definition 4.26 and Proposition 4.22 so that the ball Bk ~ B (xl:, Nl< 12) has the orbifold topological model B n IG (xl:) . The key observation in the proof of Theorem 3.24 is that we can choose a subsequence of orbifolds so that the groups G (xl:) and their actions on Bn are independent of k. We can then use the balls Bk, B k, Bk to build the limit orbifold. 4. Applications of Hamilton's compactness theorem

In this section we discuss some applications of Theorems 3.10 and 3.16. We will see more applications of the compactness theorems later in this volume. 4.1. Singularity models. Theorem 3.16 may be applied to study singular, nonsingular, and ancient solutions of the Ricci flow. For example, let (Mn,g(t)) , t E [O,T), where T E (0,00]' be a complete solution to the Ricci flow. Given a sequence of points and times {(Xk' tk)}kEN' let Kk ~ IRm (Xk' tk)l. We say that the sequence {(Xk' tk)} satisfies an injectivity radius estimate if there exists LO > 0 independent of k such that injg(tk) (Xk) ~ LoK;;I/2. Given a complete solution of the Ricci flow, we can obtain a local limit of dilations provided we have an injectivity radius estimate and a local bound on the curvatures after dilations. 3.26 (Existence of singularity models). Let (M n , 9 (t)), t E (0:, w), be a complete solution to the Ricci flow. Given a sequence of points and times {(Xk' tk)}kEN' let Kk ~ IRm (Xk' tk)1 > 0 and COROLLARY

gk (t) ~ Kkg (tk

+ K;;lt) .

Suppose that the sequence {(Xk' tk)} satisfies an injectivity radius estimate, i.e., injg(tk) (Xk) ~ LoK;;I/2,forsomeLo > 0, and suppose thatO:k,wk,O:oo,w oo

~ 0 with O:k ~ 0: > 0, Wk ~ Woo, and [tk - ~,tk +~] 00

C

(o:,w) are such

4.

APPLICATIONS OF HAMILTON'S COMPACTNESS THEOREM

that there exist positive constants p ::; (3.12) sup

00

and C <

{I~~I (x, t) : (x, t) E B9(tk) (Xk' k) x [tk -

00

143

where

; : ' tk

+ ~]} ::; CKk.

Then there exists a subsequence of the dilated solutions (B9(tk) (Xk' pK;;I/2) ,gdt) ,Xk) which converges to a solution (B~, goo (t) ,x oo ) on an open manifold on the time interval (-a oo , woo], which is complete on the closed ball Bgoo(o) (xoo,r) for all r < p. In particular, if p = 00, then the solution (Boo, goo (t), xoo) is complete. REMARK 3.27. By definition, we let B9(tk) (Xk'

00)

= M.

In Chapter 6 we shall show that the injectivity radius estimate in the corollary above for the solution (Mn, g (t)) , t E [0, T), on a closed manifold with T < 00, is a consequence of Perelman's no local collapsing theorem. THEOREM 3.28 (Local injectivity radius estimate for finite time singular solutions). Let (Mn,g(t)), t E [O,T), T < 00, be a solution to the Ricci flow on a closed Riemannian manifold. There exists fJ > such that if (xo, to) EM x [0, T) is a point and time satisfying 1 IRm (x, to)1 ::; 2" in Bg(to) (xo, p) P for some p > 0, then inj (xo) ~ fJp. g(to)

°

For finite time singular solutions on closed manifolds we have the following. COROLLARY 3.29 (Local singularity models for finite time singular solutions). Suppose (Mn, 9 (t)) , t E [0, T), T < 00, is a solution on a closed Riemannian manifold. If there exist positive constants p ::; 00 and C < 00 such that (3.12) holds, where ak, 13k ~ 0, ak - 4 a > 0, and 13k - 4 {3, then

there exists a subsequence such that (B9(tk) (Xk' pK;t/2) ,gk (t) 'Xk) converges to a solution (B~,goo (t) ,X oo ) , t E (-a,{3) , to the Ricci flow on an open manifold which is complete on the closed ball Bgoo(o) (xoo,r) for all r < p. In particular, if p = 00, then (Boo, goo (t) ,x oo ) is complete.

4.2. 3-manifolds with positive Ricci curvature revisited. As an application of Theorem 3.10 we give a proof of the following result of Hamilton, which is a consequence of Theorem 6.3 on p. 173 of Volume One.

°

THEOREM 3.30 (Closed 3d Rc > manifolds are diffeomorphic to space forms). If (M3,gO) is a closed Riemannian 3-manifold with positive Ricci curvature, then M admits a metric with positive constant sectional curvature.

144

3. THE COMPACTNESS THEOREM FOR RICCI FLOW

PROOF. Let (M, 9 (t)), t E [0, T), be the maximal solution to the Ricci flow with 9 (0) = go. Recall that T ~ ~ (Rmin (0))-1 < 00. The proof relies on three main estimates. (1) (Positivity of Ricci is preserved.) Rc (g (t)) > 0 for all t 2: O. (2) (Strong curvature pinching.) There exist C < 00 and 8 > 0 depending only on go such that

IRc -!Rgi < CR-8

R

-

on M x [0, T) (see inequality (6.37) on p. 190 of Volume One). (3) (Injectivity radius estimate.) There exists to > 0 depending on n, T, go such that if (x, t) EM x [0, T) and r E (0,1] are such that IRm (', t)1 ~ r- 2 in Bg(t) (x, r) , then injg(t) (x) 2: toT (see Theorem 3.28 or Corollary 6.62). The main technique is to dilate and apply the compactness theorem. Choose (Xk, tk) with Xk E M and tk ---t T such that

Rk ~ R (Xk' tk) =

max

M3 x [O,tk]

R 2:

max

M3 x [O,tkl

IRml

---t

00

as k ---t 00. Consider the sequence of solutions (M, gk (t) ,Xk) , t E (-tkRk' 0], where gk (t) ~ Rkg (tk + R;lt) . Let Tk ~ R~ 1/2, which is bounded above by 1 for k large enough. We have IRm (', tk)1 ~ T;2

in B9(tk) (Xk' Tk)'

Thus by (3), injg(tk) (Xk) 2: toTk, which is equivalent to injgk(o) (Xk) 2: to· Hence Hamilton's compactness theorem implies that, for a subsequence, (M, gk (t) ,Xk) converges to (M~, goo (t) ,xoo), a complete solution defined for t E (-00,0]. We claim that (Moo, goo (t)) has constant positive sectional curvature. In particular, Moo has bounded diameter and hence is diffeomorphic to M. So the theorem follows. First note that Rgoo (xoo, 0) = limk-+oo R9k (Xk' 0) = limk-+oo 1 = 1. Hence Rgoo (x,O) > 0 for x contained in a neighborhood of Xoo' Estimate (2) says that

IRc(gk) - !R(gk)gkl (t) < CR-8R( )-8 R (gk) k gk . Since the convergence of (M, gk (t) ,Xk) is in Coo on compact subsets, we have on the subset of Moo where R (goo) > 0,

IRc(gk) - !R (gk) gkl R (gk)

IRc (goo) - !R (goo) goo I

~--~~~----~---t~------~------~

R (goo)

(we have swept under the rug the fact that in Cheeger-Gromov convergence one must pull back by appropriate diffeomorphisms from an exhaustion of Moo to M; we leave it to the reader to justify the arguments in this proof).

4.

APPLICATIONS OF HAMILTON'S COMPACTNESS THEOREM

145

On the other hand, CR;;8 R (9k)-8 ----+ o· R (goo)-8 = O. Hence we conclude that Rc(goo) = kR (goo) goo on the subset of Moo where R (goo) > O. On the other hand, the contracted second Bianchi identity implies R (goo) = const in any connected subset of the set where R (goo) > O. Hence we conclude R (goo) == 1 on all of Moo, so that Rc (goo) == kgoo on Moo. 0 4.3. Ricci flow on closed surfaces with X > O. Now we give various proofs, which are variations on a theme, of the following consequence of Theorem 5.77 on p. 156 of Volume One. THEOREM 3.31. If (M 2 ,go) is a closed Riemannian surface with positive Euler characteristic, then a smooth solution 9 (t) of the Ricci flow with 9 (0) = go exists on a maximal time interval [0, T) with T < 00. Moreover, there exists a sequence {(Xk,tk)} with tk ----+ T such that gk (t) ~ Rk9 (tk + R;;1t) , with Rk = R(Xk' tk), converges to a solution (M2, goo (t)) with constant positive curvature. First we recall the ideas of some proofs from Volume One. The first proof relies on the monotonicity of the quantity lV'iV'jf - ~LlfgijI2. PROOF #1. In this proof, which is the original proof of Hamilton, we actually recall the exponential convergence (not just sequential convergence) in Coo of the normalized flow. Consider the normalized flow %t g = (r - R) 9 on the maximal time interval [0, T). In the rest of this proof we abuse natation by using 9 (t) , t E [0, T), to stand for the normalized solution rather than the unnormalized solution in the statement of the theorem. One can prove that T = 00. If R (g (to)) > 0 for some to < 00, then we may combine the entropy and Harnack (or Bernstein-Bando-Shi derivative) estimates to show that there exist c > 0 and C < 00 such that O
on M x [to, 00). Let Mij ~ V'iV'jf - ~Llfgij, where Llf ~ R - r. From the equation (see Corollary 5.35 on p. 130 of Volume One)

!!.at IMI2 = LlIMI2 - 21V'MI2 ~

2R IMI2

LlI.l\;f12 - 2c IMI2

(using the lower bound for R), we have

IMI

(3.13)

for some C1 < Volume One)

00.

~ C1 e- ct

We also have for pEN (see Corollary 5.63 on p. 149 of

(3.14)

for some Cp > 0 and Cp < 00. By the diffeomorphism invariance of the estimates (3.13) and (3.14), this implies that the modified equation

a

{)tg =

(r - R) 9 + £.'il/g

146

3.

THE COMPACTNESS THEOREM FOR RICCI FLOW

converges exponentially fast in Coo to a gradient shrinker. The nonexistence of nontrivial gradient shrinkers (Proposition 5.21 on p. 118 of Volume One) then implies that under the original equation itg = (r - R) g, the solution converges exponentially fast in Coo to a constant curvature metric. Finally, for any go we may modify the entropy estimate to show that there exists to < 00 such that R (g (to)) > o. 0 The next proof uses Hamilton's isoperimetric estimate. PROOF #IIA. Suppose that M2 is diffeomorphic to the 2-sphere. Given an embedded loop, separating M into two connected components Ml and M2, the isoperimetric ratio of, is defined by CH (r)

~ L (r)2 (Area~M1) + Area ~M2))

and the isoperimetric constant of (M, g) is CH (M,g) ~ infCH (r) ~ 47r. 'Y

Then (see Theorem 5.88 on p. 162 of Volume One) under the Ricci flow

d dt CH (M, 9 (t)) ~ 0, so that CH (M,g(t)) ~ CH (M,go)

> O.

On the other hand, in the presence of a curvature bound, the isoperimetric constant bounds the injectivity radius by inj (M,g)

~ (4~ax CH (M,9)) 1/2,

where Kmax ~ maxM K (K is the Gauss curvature). Hence we may dilate about a sequence {(Xk' tk)} approaching the singularity of the unnormalized flow 9 (t), as in the proof of Theorem 3.30, and apply Hamilton's compactness theorem to obtain a limit solution (M~, goo (t)) . This limit solution is a complete ancient solution with bounded positive curvature. In the case of a Type IIa singularity, by choosing {(Xk' tk)} suitably, the limit is an eternal solution (attaining the supremum of R in space-time), which must be the cigar soliton

(R2, t~:t!~~ ).However, the isoperimetric estimate is pre-

served in the limit,2 which contradicts the existence of such a limit. Hence the singularity is Type 1. In this case the limit (Moo,goo (t)) is compact (see [186J or Proposition 9.16 of [111]) and has constant entropy, which implies that it is a gradient shrinker and hence is a constant curvature solution. 0 2More precisely, the limit being a cigar soliton implies that limi~oo C H (M2, 9 (ti)) 0, which leads to a contradiction.

=

4.

APPLICATIONS OF HAMILTON'S COMPACTNESS THEOREM

147

REMARK 3.32. In the above proof, we could have replaced Hamilton's isoperimetric estimate by Perelman's no local collapsing theorem in Chapter 6, which enables the application of the compactness theorem and at the same time rules out the formation of the cigar soliton singularity model. Now we give a proof that Type I limits are round 2-spheres using Perelman's entropy (see Chapter 6 for properties used in the proof below; the reader may wish to come back to this part after reading that chapter). PROOF #IIB, USING PERELMAN'S ENTROPY FOR LAST STEP. For the last step in the above proof, we may use Perelman's entropy instead of Hamilton's entropy. This has the advantage that the entropy is defined for solutions with curvature changing sign, so that we may apply it to the original solution 9 (t), t E [0, T), rather than the limit solution. We assume that 9 (t) forms a Type I singularity. Let W(g (t), f (t), T (t)) denote the entropy with T ~ T - t, which is defined for T E (0, T]. Taking f to be the constant h (t) ~ -log vOl~7)(M)' so that it satisfies the constraint

f M (47rT)-l e- h dJ.1 = 1, we see that J.1 (g (t) ,T (t)) ::; W(g,

47rT

h, T) ::; TRmllJ{ (t) - log Volg(t) (M) - 2.

In particular, by the long-time existence theorem for the Ricci flow on the 2-sphere, we have Volg(t) (M) = 87r (T - t). Hence we have an upper bound for the J.1-invariant: J.1 (g (t) , T (t)) ::; C - 2 + log 2,

where we have used the Type I assumption (T - t) RmllJ{ (t) ::; C. In particular, by the monotonicity of J.1, 1£ J.1 (g (t) , T (t)) ~ 0, the limit J.1T ~ lim J.1 (g (t) , T (t) ) t-+T

exists. Dilate the solution about (Xk' tk) with gk (t) = Rkg (tk + R"k1t) as in the proof of Theorem 3.30. By the scaling property of J.1 we have J.1 (gk (t) ,Rk (T - tk) - t)

= J.1 (g (tk + R"k1t) ,T - tk - R"k 1t ) .

Thus for each t E (-00, woo), the (maximal) time interval of existence of the limit solution (M~,goo (t)) , J.1T

== lim J.1 (gdt) ,RdT - tk) - t) = J.1 (goo (t) ,woo - t) . k-+oo

(We may assume Rk (T - tk) ~ Woo converges. Here we also used the continuity of J.1 (g, T) in 9 and the fact that the convergence, after the pull-back by diffeomorphisms, of gk (t) to goo (t) is globally pointwise in Coo since M~ ~ M is compact.) Now the theorem follows from the result that a solution having constant J.1 is a gradient shrinker. D

148

3. THE COMPACTNESS THEOREM FOR RICCI FLOW

5. Notes and commentary

Some basic references for compactness theorems for Riemannian metrics are Cheeger [70], Greene and Wu [165], Gromov [169], and Peters [300]. A survey of compactness theorems in Riemannian geometry has been given in Petersen [301]. The compactness theorem for Ricci flow in this chapter was proven by Hamilton in [187] and was used to classify singularities and nonsingular solutions in [186], [190] and [297]. Cheeger-Gromov theory was also directly used to study the Ricci flow in Carfora and Marzuoli [57]. Further compactness theorems on the Ricci flow which extend Hamilton's results can be found in [257] and [156]. It should be noted that there has been much work to ensure the injectivity radius bound for dilations of singularities, most notably by Hamilton [186] and Perelman [297]. Additional work on injectivity radius estimates has been done by Wu [372] and by the authors of [112] in the case of 2-dimensional orbifolds and [109] for sequences of solutions with almost nonnegative curvature operator.

CHAPTER 4

Proof of the Compactness Theorem We think in generalities, but we live in details. - Alfred North Whitehead There is no royal road to geometry. - Euclid

1. Outline of the proof

We now prove the compactness Theorem 3.9. This is a fundamental result in Riemannian geometry and does not require the Ricci flow. The compactness theorem is in the spirit of Cheeger [70] and Gromov [169] (see also Greene and Wu [165]' Peters [300], and the book [37]). We follow the proof for pointed sequences converging in Coo given by Hamilton [187]; as Hamilton notes there, things are easier because we can assume bounds on all covariant derivatives of the curvature. Theorem 3.9 will be proved in several steps. It is outlined as follows. STEP A: Construct a sequence of coverings of each manifold Mk which we can compare to each other. The covers should consist of balls BI: C Mk with a number of properties, most notably: • they are diffeomorphic to Euclidean balls, and for each fixed O! they have the same radii for all sufficiently large k, • they are numbered sequentially in O! starting from balls centered at the origin to balls with centers further and further away from the origin, • if we take smaller radii (BI: c BI:), they are disjoint, and if we take larger radii (BI: C BI: c BI:), they contain their neighbors, • we can bound the number of balls intersecting a given ball (the most is I (n, Co), where Co is the curvature bound), and • we can bound the number of these balls that it takes to cover a large ball in Mk, independent of k for k large (it takes fewer than A (r) balls to cover B (Ok, r) if k 2': K (r)). The specifics of this are contained in Lemma 4.18. This process is carried out in Section 3 of this chapter. STEP B: Use our nice covering to construct maps Ffd : BI: ~ M£. We do this by taking the inverse of the exponential map on the ball BI: C Mk, identifying the tangent space of Mk at the center of the ball BI: with Euclidean space and with the tangent space of M£ at the center of B't and 149

4.

150

PROOF OF THE COMPACTNESS THEOREM

then mapping by the exponential map to Me. We do this for each ball. This is done in subsection 4.1 below. STEP C: Use nonlinear averaging to glue together the maps Ffj to obtain maps Fke : B (Ok, 2k) --+ Me which take Ok to Oe (done in subsection 4.2 below). By taking subsequences, we can ensure that compositions of Fk,k+1 are approximate isometries which are getting closer to isometries as k goes to 00. STEP D: We form the limit manifold M~ as the direct limit of the directed system {Fk,k+1 : B (Ok, 2k) --+ B (Ok+1, 2k+l)} . The coordinates of B (Ok, 2k) then form coordinates for the limit Moo, i.e., for each coordinate Hk : EO: --+ B (Ok, 2k) there is a coordinate for the limit manifold defined as H~,k = h 0 Hk : EO: --+ Moo, where Ik is the inclusion of B (Ok, 2k) into Moo. Furthermore, for each coordinate EO: of B (Ok, 2k) there are Riemannian metrics 9k,e which are obtained by the pullbacks F"kege. Since Fke are approximate isometries, the sequence is equicontinuous and thus by the Arzela-Ascoli theorem we can find a convergent subsequence and get local metrics g~ k' It is not hard to see that these metrics form a Riemannian metric 900' on Moo via the coordinate charts H~ k . We can then show that the limit metric is complete and (Moo, 900' 0 00 ) s~tisfies the theorem, where 0 00 is the equivalence class of the base points in the direct limit. This is done in subsection 4.4 below. At many points in this construction we will take a subsequence; to simplify notation, at each stage the sequence will be re-indexed to continue to be k. 2. Approximate isometries, compactness of maps, and direct limits In this section we shall introduce some basic concepts which are essential to the construction of the limit manifold (M~, 900,000 ) . 2.1. Approximate isometries. In the following, the notation ITlg means the length at a point of the tensor T with respect to the Riemannian metric 9. DEFINITION 4.1 (Approximate isometry). For any 0 < c < 1 and p E NU {O}, a smooth map {[> : (M n ,9) --+ (Nn, h) is an (c,p)-pre-approximate isometry if sup I{[>*h - 91g ~ c, xEM

sup sup 1\7~ ({[>*h) l:'So::'SpxEM

I

~ c.

g

An (c,p)-pre-approximate isometry is an (c,p)-approximate isometry if it is a diffeomorphism and

I

sup ({[>-1)* g - hi xEN

h

~ c,

2.

APPROXIMATE ISOMETRIES, COMPACTNESS OF MAPS, DIRECT LIMITS 151

i.e., <1>-1 :

(N, h)

~ (M,

g) is also an (E,p)-pre-approximate isometry.

Note the condition I(<1>-1)* 9 - hlh ~ 10 is equivalent to Ig - <1>*hlcp*h ~ 10 and IVI: [(<1>-1)* g] Ih ~ 10 is equivalent to IV~*hglcp*h ~ E. Another way to express the condition sUPXEM 1<1>*h - gig ~ 10 is

j)

o<1>ao<1>b 12 _ ([ ik O<1>a] o<1>b k) (O<1>C [ .e O<1>d] 1hab oxi oxj - gij 9 habg oxi oxj - I j oxk hcdi' oxf. - I k =

I(d<1»T (d<1»

- idl2 ,

where id is the identity map on T M, the transpose comes from the two metrics hand g, and 1F12 = trace (F2) . Approximate isometries allow pointwise bounding of metric tensors as follows. PROPOSITION 4.2 (Approximate isometries and norms). Let 10 E (D, 1). (i) If<1>: (Mn,g) ~ (Nn,h) is an (E,D)-pre-approximate isometry and X is a vector field on M, then

IXI!*h ~

(1 + E) IXI~. (ii) If <1> : (Mn,g) ~ (Nn,h) is an (10, D)-approximate isometry and X is a vector field on M, then 1 2 2 2 1 + 10 IXlcp*h ~ IXl g ~ (1 + E) IXlcp*h' PROOF. (i) Using the Cauchy-Schwarz inequality on the tensor space, we find that

(<1>*h)ij XiXj

= ((<1>*h)ij - 9ij) XiXj + gijXiXj ~ 1<1>*h - gig' gijXiXj

+ gijXiXj

+ E) gijXiX j . (1 + E) IXI!*h follows ~ (1

(ii) Inequality IXI~ ~ from (i) and the fact that <1> -1 : (N, h) ~ (M, g) is also an (10, p)- pre-approximate isometry. 0 The following now immediately follows from Lemma 3.13. COROLLARY 4.3 (Norms of tensors). If <1> : (Mn, g) ~ (Nn, h) is an

(10, D)-approximate isometry, then for any (p, q)-tensor field T on M we have (4.1) (1 + E)-(P+q)/2ITlcp*h ~ ITlg ~ (1 + E)(P+Q)/2ITlcp*h. The following proposition shows how (10, D)-approximate isometries deform distances by small amounts. PROPOSITION 4.4 (Distances). If <1> : (M n , g) ~ (Nn, h) is an (10, D)-

pre-approximate isometry, then <1> (Bg (xo, r)) C Bh (<1> (xo) , (1

+ 10)1/2 r)

.

152

4. PROOF OF THE COMPACTNESS THEOREM

x

PROOF. Suppose

E

Bg (xo, r). Then

dh (