1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

4. APPLICATIONS OF HAMILTON'S COMPACTNESS THEOREM 143

that there exist positive constants p :::; oo and C < oo where

(3.12)

sup {1~~1 (x, t): (x, t) E Bg(tk) (xk, k) x [tk - ;: , tk + ;:] } :::; CKk.

Then there exists a subsequence of the dilated solutions

( Bg(tk) (xk, pKk

1

!

2

) , gk(t), Xk)

which converges to a solution (B~, g 00 (t), x 00 ) on an open manifold on the

time interval (-a 00 , w 00 ], which is complete on the closed ball B 900 (o) (x 00 , r)

for all r < p. In particular, if p = oo, then the solution (B 00 , g 00 (t), x 00 ) is

complete.

REMARK 3.27. By definition, we let Bg(tk) (xk, oo) = M.

In Chapter 6 we shall show that the injectivity radius estimate in the

corollary above for the solution (Mn, g (t)), t E [O, T), on a closed manifold

with T < oo, is a consequence of Perelman's no local collapsing theorem.

THEOREM 3.28 (Local injectivity radius estimate for finite time singular

solutions). Let (Mn,g(t)), t E [O,T), T < oo, be a solution to the Ricci

flow on a closed Riemannian manifold. There exists 8 > 0 such that if

(xo, to) EM x [O, T) is a point and time satisfying

for some p > 0, then

1

IRm (x, to)I :S (^2) p in Bg(to) (xo, p)
inj (xo) ?:: 8p.
g(to)
For finite time singular solutions on closed manifolds we have the fol-
lowing.
COROLLARY 3.29 (Local singularity models for finite time singular so-

lutions). Suppose (Mn, g (t)), t E [O, T), T < oo, is a solution on a closed

Riemannian manifold. If there exist positive constants p :::; oo and C < oo

such that (3.12) holds, where ak, !A ?:: 0, ak ---+ a > 0, and f3k ---+ /3, then

there exists a subsequence such that ( Bg(tk) ( Xk, pKk^112 ) , gk (t), Xk) con-

verges to a solution (B~, goo (t), Xoo), t E (-a, /3), to the Ricci flow on
an open manifold which is complete on the closed ball B 900 (o) (x 00 , r) for all
r < p. In particular, if p = oo, then (B 00 , g 00 (t), x 00 ) is complete.

4.2. 3-rnanifolds with positive Ricci curvature revisited. As an
application of Theorem 3.10 we give a proof of the following result of Hamil-
ton, which is a consequence of Theorem 6.3 on p. 173 of Volume One.

THEOREM 3.30 (Closed 3d Re> 0 manifolds are diffeomorphic to space

forms). If (M^3 ,g 0 ) is a closed Riemannian 3-manifold with positive Ricci

curvature, then M admits a metric with positive constant sectional curva-

ture.