1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

4. DERIVATIVE ESTIMATES AND SOME CONJECTURES 149

with

(20.47)

The limit solution ( M~, 900 ( t)) is complete with nonnegative curvature

operator and is ,,;-noncollapsed at all scales. On the other hand, from

R-gk (xk, 0) = R9k (xk, 0)5k = bk we have that

R_g°" (x 00 , 0) = 0

and hence by Hamilton's strong maximum principle (M 00 ,9 00 (t)) is a fl.at

solution. Since (M 00 ,9 00 (t)) is fl.at, it follows from ,,;-noncollapsing that

(M 00 , 900 (t)) has maximum volume growth, i.e.,

Vol_g=(O) B9=(o)(x 00 , r) 2: Krn

for all r > 0. Since a complete fl.at metric with maximum volume growth

must be Euclidean space, this contradicts (20.47). The claim is proved.

For any r > 1, we have

Vol 9 k(o) B-gk(o)(xk, 4r) 2: Vol 9 k(o) B 9 k(o)(xk, 1)

co ( )n

2: co = ( 4 r t 4r.

Applying Proposition 20.6(b) to Bgk(O) (xk,4r), we get for z E Bgk(O) (xk,r)
and r 2: 1,
C (£Q_) B (_£Q)
R-(z, 0) < (4r)n + (4r)n.
9k - 16r2 l6r2ro (£Q_)
(4r)n
This and the K-noncollapsing assumption enable us to apply Hamilton's
compactness theorem and we get a subsequence (Mk, gk (t), xk) converging

to (M~, g 00 (t), x 00 ). Clearly the limit is a ,,;-solution with Harnack and

Theorem 20.9 is proved. D

4. Derivative estimates and some conjectures

It is remarkable that compactness can be used to obtain derivative esti-
mates rather than the other way around.

4.1. Derivative estimates for ,,;-solutions.
A simple consequence of Perelman's compactness theorem, i.e., Theorem
20.9, is the following derivative estimate for ,,;-solutions with Harnack.

THEOREM 20.17 (Derivative estimates for K-solutions satisfying trace

Harnack). Given K > 0 and n 2: · 2, there exists a constant rJ ( K, n) < oo

such that for any K-solution (Mn, g (t)), t ::; 0, with Harnack we have the

estimates

(20.48) IV' R (x, t) I ::; 'r/ (K, n) R (x, t)^312 , I~~ (x, t) I ::; 'rJ (K, n) R (x, t)^2

for any x EM ·and t::; 0.