1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

146 3. THE COMPACTNESS THEOREM FOR RICCI FLOW

converges exponentially fast in C^00 to a gradient shrinker. The nonexistence
of nontrivial gradient shrinkers (Proposition 5.21 on p. 118 of Volume One)
then implies that under the original equation gt9 = (r - R) 9, the solution
converges exponentially fast in C^00 to a constant curvature metric. Finally,
for any 90 we may modify the entropy estimate to show that there exists

t 0 < oo such that R (9 (to)) > 0. 0

The next proof uses Hamilton's isoperimetric estimate.

PROOF #IIA. Suppose that M^2 is diffeomorphic to the 2-sphere. Given
an embedded loop 'Y separating Minto two connected components Mi and

M2, the isoperimetric ratio of/ is defined by

CH(r)-=;:.L(r)2 (Area~M1) + Area~M2))

and the isoperimetric constant of ( M, 9) is

CH (M, 9)-=;:. inf CH (r) :S 47r.

'Y

Then (see Theorem 5.88 on p. 162 of Volume One) under the Ricci :flow

d

dt CH (M, 9 (t)) 2: 0,

so that

CH (M,9 (t)) 2: CH (M,90) > 0.

On the other hand, in the presence of a curvature bound, the isoperimetric

constant bounds the injectivity radius by

inj(M, 9) 2: ( 4 ~ax CH (M, 9))

112

,

where Kmax-=;:. maxM K (K is the Gauss curvature). Hence we may dilate
about a sequence { ( x k, tk)} approaching the singularity of the unnormalized
flow 9 (t), as in the proof of Theorem 3.30, and apply Hamilton's compact-
ness theorem to obtain a limit solution (M~, 900 (t)). This limit solution is
a complete ancient solution with bounded positive curvature. In the case of
a Type Ila singularity, by choosing {(xk, tk)} suitably, the limit is an eter-
nal solution (attaining the supremum of R in space-time), which must be

the cigar soliton ( JR.^2 , f~:t!t~). However, the isoperimetric estimate is pre-

served in the limit,^2 which contradicts the existence of such a limit. Hence
the singularity is Type I. In this case the limit (M 00 ,9 00 (t)) is compact (see
[186] or Proposition 9.16 of [111]) and has constant entropy, which implies
that it is a gradient shrinker and hence is a constant curvature solution. 0

(^2) More precisely, the limit being a cigar soliton implies that limi--+oo CH ( M (^2) , g (ti)) =
0, which leads to a contradiction.