1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

130 3. THE COMPACTNESS THEOREM FOR RICCI FLOW

DEFINITION 3.6 (C^00 -convergence of solutions after diffeomorphisms).

A sequence {(Mk, gk ( t) , Ok)}kEN , t E (a, w) , of complete pointed solutions

to the Ricci flow converges to a complete pointed solution to the Ricci flow

(M~, 900 (t), 000 ), t E (a, w), if there exist

(1) an exhaustion {UkhEN of M 00 by open sets with Ooo E Uk, and

(2) a sequence of diffeomorphisms cI>k : Uk -+ Vk ~ cI>k (Uk) c Mk with

cI>k ( Ooo) = Ok

such that (Uk, cI>l:: [ gk (t)ivk]) converges in C^00 to (Moo, 900 (t)) uniformly

on compact sets in M 00 x (a, w).

REMARK 3. 7. The last statement in the above definition is a slight abuse

of notation; we really mean (Uk x (a, w), cI>l:: [ 9k ( t) I vk J + dt^2 ) converges in

C^00 to (M 00 x (a, w),9 00 (t)+dt^2 ) uniformly on compact sets in M 00 x (a,w)

using Definition 3.2, where dt^2 is the standard metric on (a, w).

When there is a bound on the curvatures (recall that when given a sin-
gular solution and a suitable sequence of space-time points, the choice of
dilation factors is chosen to guarantee this for the associated sequence of
solutions) and an injectivity radius estimate for a sequence of solutions to
the Ricci flow, then the following compactness theorem provides a subse-
quence which will converge in the C^00 -Cheeger-Gromov sense. We end this
subsection with a definition which is related to the assumption of bounded
curvature.

DEFINITION 3.8 (Bounded geometry). We say that a sequence or family

of Riemannian manifolds has bounded geometry if there exist positive

constants Gp such that

j\7P Rml :S Gp

for all p E NU {O} and for all metrics in this sequence or family. That is,
the curvatures and their covariant derivatives of each order have uniform
bounds.

1.2. Statements of the compactness theorems. Let injg (0) de-

note the injectivity radius of the metric 9 at the point 0. For sequences of
Riemannian manifolds we have the following convergence theorems ( origi-
nally proven in [187]).

THEOREM 3.9 (Compactness for metrics). Let {(Mk,9k, Ok)}kEN be a

sequence of complete pointed Riemannian manifolds that satisfy

(1) (uniformly bounded geometry)

/\7f Rmk/k :S Gp on Mk

for all p ~ 0 and k where Gp < oo is a sequence of constants

independent of k and