1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

1. COMPACT FINITE TIME SINGULARITY MODELS ARE SHRINKERS 13

PROOF. Given TE [c-^1 ,CJ, let w with JMw^2 dμ 9 = 1 be a minimizer

of the entropy K (g, ·, T) in (17.7). Note that by (17.20) and assumptions

(1) and (2),

μ (g' T) = K (g' w' T)

:s; T Ravg (g) +log Vol (g) + ~log C

:s; canst ( n, C)

for T E [ c-^1 , CJ. Define c E IR+ so that

JM (cw)

2

dμ 9 = 1;

we may make c arbitrarily close to 1 by choosing o sufficiently small. We

have

μ (9, T) :s; K (9, cw, T)

=JM (T ( R 9 (cw)

2

+ 4 l\7 (cw) I;) -(log ((cw)

2

)) (cw)^2 ) dμ 9

n

• 2

1og(47rT) - n

:s; μ (g, T) + T JM w^2 ( c^2 Rgdμg - R9dμ9)

• 
  

  4T JM ( c^2 l\7wl; dμg - l\7wl~ dμ9)

  
  


• 
  

  JM w^2 log ( w^2 ) ( dμ9 - c^2 dμ 9 ) -c^2 log ( c^2 ) JM w^2 dμ 9

  
  

since K (g, w, T) = μ (g, T). Thus for any c > 0, by taking o sufficiently small
in assumptions (3) and (4) and by making c sufficiently close enough to 1,
we obtain
μ (9, T) - μ (g, T) :s; c.

Here we used the fact that the logarithmic Sobolev inequality implies that

JM l\7wl~dμ9 and JM w

2

log (w

2

) dμ 9

are bounded byμ (g, T) +canst (n, C), which in turn is uniformly bounded

(see the proof of Lemma 6.24 in Part I or (17.58) below). D

In dimension 3 any shrinking gradient Ricci soliton on a closed 3-manifold
is a constant positive sectional curvature solution (see Theorem 9.79 in [45]
for example; note that compact quotients of S^2 x IR cannot be Kr noncollapsed
at scales), so that we have the following.

COROLLARY 17.16 (Singularity models on closed 3-manifolds are round).

If (M~, 900 (t)), t E (-oo, OJ, is a finite time singularity model on a closed

3-manifold, then 900 (t) is a shrinking spherical space form.