112 2. KAHLER-RICCI FLOW
by taking 'Y (t) to be a minimal geodesic, with respect to g (ti), joining x1
to x2 with speed l'Y(t)i 9 (ti) = ce-;;(t-ti), where
Thus
(2.116)
R (x2, t2) > 1 - e--ii:ti exp (-_!__ (l - e--ii;(t 2 -t1))-l d2 (x x )).
R (x1, ti) - 1-e--ii;t2 4n g(t1)^1 '^2
Note by the inequality x ::::; ex - 1 that we have
----R(x2,t2) > l-e--ii;t1 exp ( --en^1 I:.(t2-t1)d;(t1)(x1,x2))
R (x1, tl) - 1 - e--ii:t^2 4 t2 - ti '
which is the estimate we would have obtained from (2.115) by using e-ii:(t-ti) ::::;
e-ii:(t^2 -t^1 ) and taking 'Y (t) to be a minimal geodesic joining x 1 to x2 with speed
I
. (t)J _ d 9 (ti)(x1,x2)
'Y g(ti) - 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 fl.ow 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 fl.ow on closed
manifolds with positive bisectional curvature (Theorem 2.70).
THEOREM 2.92 (NKRF: Kee (V, W) 2:: 0 curvature estimate). IJ(Mn,go)is a closed Kahler manifold with ~ [wo] = 27rq (M) and nonnegative bisec-
tional curvature, then the solution g (t) to the normalized Kahler-Ricci flow,
with g ( 0) = go, has uniformly bounded curvature for all time.
PROOF. Without loss of generality we may assume the solution is nonfl.at
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 EM
n = R(y,t+l) > 1-e-t ex (-d;(t)(x,y))
R (x, t) R (x, t) - 1 - e-(t+l) p 4 (1 - cl) ·S. 1-e-(t+i)^1 e-
(^2 1)