1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

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 g is Kahler-Einstein if p =.Aw for some

.A ER If Mn admits a Kahler-Einstein metric g, then

(We have .A = ~' where the (complex) scalar curvature R is constant.)

Therefore, a necessary condition for the existence of a Kahler-Einstein met-

ric on M is that its first Chern class have a sign. By having a sign we mean

that c1 (M) = 0, < 0, or > 0 if there exists a real (1, 1)-form in the first

Chern class which is zero, negative definite, or positive definite, respectively.

COROLLARY 2.29 ( c 1 = 0: existence of Kahler Ricci fl.at metrics). If

(Mn, go) is a closed Kahler manifold with ci (M) = 0, ·then there exists a

Kahler metric g with [w] = [w 0 ] such that Re (g) = 0.

Kahler Ricci fl.at metrics are called Calabi-Yau metrics and Kahler

manifolds with c1 (M) = 0 are called Calabi-Yau manifolds. Another

consequence of Theorem 2.28 is that if c 1 .(M) < 0 (respectively, c 1 (M) >

0), then in each Kahler class there exists a metric with negative (respectively,

positive) Ricci curvature.

When c1 (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 (ci < 0 Calabi conjecture: R < 0 Kahler-Einstein met-

rics). If Mn is a closed complex manifold with c1 (M) < 0, then there exists

a Kahler-Einstein metric g on M, which is unique up to homothety (scal-

ing), with negative scalar curvature.

A consequence of Theorem 2.30 is the following Chern number inequal-

ity:

(-lt c1(Mt :S (-lt

2

(n + l) c1(Mt-^2 c2(M).

n

(See Yau [378] and also Corollary 9.6 on p. 226 of [383] for an exposition.)

REMARK 2.31. Another consequence of Theorem 2.30 is that if (M^2 , g)
is a closed Kahler surface homotopically equivalent to CP^2 , then M^2 is
biholomorphic to CP^2 (see Yau [378]; earlier related work in all dimensions
was done by Hirzebruch and Kodaira [204]).

If c1 (M) > 0, however, there are obstructions to the existence of Kahler-
Einstein metrics. An example is the Futaki invariant (see [147]). On a

closed manifold Mn with c1 > 0 (i.e., a Fano manifold), fix a Kahler metric

g such that [w] is a positive real multiple of c1(M). (This is the so-called