1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

150 20. COMPACTNESS OF THE SPACE OF 1;;-SOLUTIONS

PROOF. If the theorem is not true, then for some K > 0 and n 2:: 2

there exist a sequence of K-solutions (Mk,gk(t)), t::; 0, with Harnack and

space~time points (xk, tk) E Mk x (-oo, OJ such that we have either

IV' Rgk (xk, tk)I 2:: kRgk (xk, tk)^312

for all k 2:: 1 or

I gt Rgk (xk, tk) 12:: k · Rgk (xk, tk)

2

for all k 2:: 1. Define the rescaled solutions (Mk,gk(t),xk), t E (-oo,O], by

gk(t) = Rgk(xk, tk) · 9k (tk + R ( t t )) ·

gk Xk, k

Then Rgk(o)(xk) = 1 and we have either

['VR:gk (xk,0)[ 2:: k for all k or 1:tR9k(xk,O)l 2:: k for all k.

On the other hand, by Theorem 20.9 there exists a subsequence

(Mk, gk(t), Xk)-+ (M~,goo(t), Xoo)

converging in the Cheeger-Gromov 000 -topology. In particular,

['V R:gk (xk, 0) [ -+ ['V Rg 00 (xoo, 0) [

and

a a

at R:gk (xk, 0) -+ at R9 00 (x^00 , 0).

It is impossible to have both of these convergences since either the sequence

['V R:gk (xk, 0) [ or the sequence [ gtR9k (xk, 0) [ is divergent. The theorem is

proved. D

REMARK 20.18 (Estimates for derivatives of any order). It is clear that

the above proof also implies the following. For any K > O, f, m, and n E

N - { 1}, there exists a constant rJ ( K, f,, m, n) < oo such that for any K-

solution (Mn, g ( t)), t ::; 0, with Harnack we have the estimates

(20.49) I ::e \lm R (x, t) I ::; rJ (K, f, m, n) R (x, t)l+f!+2¥'

for any x E Alf. and t::; 0. (The same result holds for compositions of gt and

\7, in any order, of either Rm, Re, or R with the form of the RHS unchanged.)

Note that estimate (20.49) is scale invariant, that is, if g satisfies (20.49),

then C · g satisfies (20.49) for any CE (0, oo).

Combining Theorem 19.56 and Theorem 20.17, we obtain the following.

COROLLARY 20.19 (Universal scaled derivative bounds in dimension 3).

There exists a universal constant rJo < oo (independent of K) such that for

any 3-dimensional K-solution ( M^3 , g ( t)), t ::; 0, we have the estimates

(20.50) l'VR(x,t)I::; rJoR(x,t)~, I~~ (x,t)I::; rJoR(x,t)^2