- MATRIX DIFFERENTIAL HARNACK ESTIMATE 113
In particular, if x E Bt (y, 1) , thenR (x, t) :::; n (1 + e-^1 ) exp ( 4 (l ~ e-l)).
Hence, since g (t) has nonnegative bisectional curvature, the curvature of
g (t) is bounded^9 in Bg(t) (y, 1). By Perelman's no local collapsing theo-
rem, which holds for a solution to the normalized Kahler-Ricci fl.ow since
its statement is scale-invariant and since the corresponding solution to the
Kahler-Ricci fl.ow 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: /'i,.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 EM be a point with d (x, y) = d 2: 2. Since Re 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 + 1r - (d - 1r
Volg(t)(Bg(t)(x,d-1)) :::; (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))
Volg(t) (Bg(t) (y,1))
2: C(n) d.Taking d = diamg(t) (M) , we have
. ( C (n) Volg(t) (M) C (n) Volg(t) (M)
diamg(t) M) < < -----'--'-~-
- Volg(t) (Bg(t) (y, 1)) - /'i,
Since under the normalized fl.ow, 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):::; n (1 + e-
1
) exp ( 4 (l ~2
e-l)),which is our desired uniform estimate for R. D
9We actually only need an upper bound on the scalar curvature for Perelman's :h.o
local collapsing theorem (Theorem 6.74).