1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

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

llfllc2,a(M) :S C2e-t

for some C2 < oo. Iterating the Schauder estimate, we have llfllc2m,<>(M) ::::;

Cme-t for some constants Cm< oo and all m EN. This implies the estimate

II

~ II 02m,<>(M) :S C2e-t and hence implies the exponential convergence of

cp ( ·, t) ----+ cp 00 ( ·) in C^00 as t ----+ oo for some smooth function cp 00 • This proves

that the normalized Kahler-Ricci fl.ow converges in C^00 to a Kahler-Einstein

metric with negative scalar curvature. Theorem 2.51 is proved.

6. Survey of some results for the Kahler-Ricci flow.

6.1. Closed Kahler manifolds with nonnegative bisectional cur-

vature. Using the short-time existence of the Kahler-Ricci fl.ow, the result
of Mori, Siu and Yau (Theorem 2.33) was generalized by Banda [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 curva-

ture). If (Mn,g) is a closed Kahler manifold with nonnegative bisectional

curvature, then its universal cover (Mn, g) is isometrically biholomorphic to

the product of complex Euclidean space, compact irreducible Hermitian sym-

metric 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 sec-
tional curvature (i.e., the Fubini-Study metric).

The proof of the theorem above uses the following result, proved by

Banda 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 (0)) is a closed Kahler manifold with nonnegative bisectional cur-

vature, then the solution g (t) to the Kahler-Ricci flow has nonnegative bi-

sectional curvature for all t ?: 0. If in addition g (0) has positive Ricci cur-

vature at one point, then g (t) has positive holomorphic sectional curvature

and positive Ricci curvature for all t > 0.

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].

(^7) The case of nonnegative curvature operator was considered by Cao and one of the
authors [50].