1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

10 17. ENTROPY, μ-INVARIANT, AND FINITE TIME SINGULARITIES

On the other hand, it is possible for a finite time singular solution on a

closed manifold that .>.. (g (t)) > 0 for all t E [O, T) and that all singularity

models are noncompact. Such an example on sn is given by Angenent and
one of the authors [7], where a finite time neckpinch (singularity model is
sn-l x IB.) is exhibited for a class of rotationally symmetric solutions with

R (g (t)) > 0 (which implies.>.. (g (t)) > 0).

1.4. Classification of compact finite time singularity models.
In this subsection we discuss the following application of bounds for the
μ-invariant to the classification of compact finite time singularity models as
shrinking gradient Ricci solitons by Z.-L. Zhang (see Theorem 1.1 in [197]).
In the a priori special case of singularity models of Type I singular solutions,
this result was proved earlier by Sesum [169].

THEOREM 17.13 (Compact finite time singularity models are shrinkers).

If (M;;, g 00 (t)), t E (-oo, OJ, is a finite time singularity model, where M 00

is a closed manifold, then g= (t) is a shrinking gradient Ricci soliton.

PROOF. STEP 1. limt-+T v (g (t)) exists. By assumption, there exists a

singular solution to the Ricci fl.ow on a closed manifold (Mn, g (t)), t E [O, T),

where T < oo, and there exists a sequence (xi, ti) with ti -+ T such that

(17.29) (Mn, Qig (ti+ Qi^1 t)) -+ (M;;, g= (t))

in the sense of C^00 Cheeger-Gromov convergence fort E (-co, OJ and for

Qi ~ JRmJ (xi, ti) -+ oo.

By Corollary 17.12, we may assume by translating time that

)\ (g (t)) > 0

for all t 2: 0. By

(17.30) -()() < l/ (g ( t)) < 0

(see Remark 17.7(2) above and (17.47) and (17.42) below) and the mono-
tonicity of v (g (t)) (see Lemma 6.35(1) in Part I), we have that

(17.31) VT~ limv(g(t)) E (-oo,O]
t-+T
exists.

STEP 2. .>.. (g 00 (t)) > 0. Since M 00 is compact, M 00 is diffeomorphic to

M and there exist diffeomorphisms I.pi : M 00 -+ M such that

cpt ( Qig (ti+ Qi^1 t)) -+goo (t)

converges pointwise in Ck on M 00 x [-k, O] for each k E N. Hence, by

Lemma 5.24 in Part I,

A (goo (t)) = ,lim A (cpt (Qig (ti+ Qi^1 t)))

i-+oo

= i~1! Qil .>.. (g (ti+ Q-;lt))

2: 0.