1547671870-The_Ricci_Flow__Chow

3. THE COMPACTNESS THEOREM 231

for 0 :S: t :S: a/ K , where Cm~ J /3m (m - 1)! +a (Cm+ f3C:n) depends only

on m, n, and max {a, 1}. Thus we have

1 '7mR v m< I - a -tm <- --CmK tm/ 2

a

for 0 < t ::::; K.

This completes the inductive step, hence our proof of Theorem 7.1. D

3. The Compactness Theorem
   The Compactness Theorem for the Ricci fl.ow [64] tells us that any
   sequence of complete solutions to the Ricci fl.ow having curvatures uniformly
   bounded from above and below and injectivity radii uniformly bounded from
   below contains a convergent subsequence. This result has its roots in the
   convergence theory developed by Cheeger and Gromov. In many contexts
   where this theory is applied, regularity is a crucial issue. By contrast, the
   proof of the Compactness Theorem for the Ricci fl.ow is greatly aided by
   the fact that sequences of solutions to the Ricci fl.ow enjoy excellent regu-
   larity properties. (See Corollary 7.3 above.) Indeed, it is precisely because
   bounds on the curvature of a solution to the Ricci fl.ow imply bounds on
   all derivatives of the curvature that the compactness theorem produces C^00
   convergence on compact sets. For this reason, we state it in this chapter.
   We plan to present a detailed proof in a successor to this volume.
   The most important application of the compactness theorem is the for-
   mation of singularity models (final time limit flows). These are complete
   nonfl.at solutions of the Ricci fl.ow that occur as limits of sequences of di-
   lations about a singularity. Indeed, we already used it for this purpose in
   Section 15 of Chapter 5. We shall discuss singularity models in greater
   generality in Chapter 8.
   THEOREM 7.8 (Compactness Theorem). Let

{Mi,gi (t), Oi, Fi: i EN}

be a sequence of complete solutions to the Ricci flow existing for t E (a, w),

where -oo ::::; a < 0 < w ::::; oo. Each solution is marked by an origin

Oi E Mi and a frame Fi = { ei, ... , e~} at Oi which is orthonormal with

respect to 9i ( 0). Suppose that there exists K < oo such that the sectional

curvatures of the sequence are uniformly bounded by K in the sense that

sup !sect [gi] I :S: K

Mjx(a,w)

for all i E N. Suppose further that there exists 6 > 0 such that the injectivity

radii of the sequence are bounded at Oi E Mi and t = 0 in the sense that

inj g;(O) ( Oi) 2: 6

for all i E N. Then there exists a subsequence which converges in the pointed

category to a complete solution

{M~, 900 (t), Ooo, Foo}