1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

1. COMPACT FINITE TIME SINGULARITY MODELS ARE SHRINKERS 9

REMARK 17.9. For an elementary upper bound of the volume of a solu-

tion when)..> O, see the notes and commentary at the end of this chapter.

Recall that a complete (finite time) singularity model (M~, g 00 (t)),

t E (-oo, OJ, is obtained from taking the limit of rescalings of a finite time

singular solution of the Ricci flow (Mn, g (t)), t E [O, T), on a closed manifold

(see §1 in Chapter 19 in this volume and Chapter 8 of [45]).

The volume lower bound has the following consequence for singularity

models associated to solutions with)..:::::; O; this is Corollary 3.2 in [197].

LEMMA 17.10 (Finite time singularity models are noncompact when

).. :::::; 0). If (Mn, g (t)), t E [O, T), where T < oo, is a singular solution

to the Ricci flow on a closed manifold with ).. (g (t)) :::::; 0 for all t E [O, T),
then any corresponding singularity model is noncompact.
PROOF. This follows since by (17.28) there exists c > 0 such that
Vol (g (t)) ;::::: c for all t E [O, T), whereas for any blow-up sequence the
dilation factors tend to infinity. More explicitly, suppose that ti /" T and
Pi E M are such that Ki ~ JRm g(ti) (Pi) J ----+ oo and suppose that the se-
quence (Mn,gi(t),pi), where

9i ( t) ~ Ki · g (ti + ;J ,

converges to a complete ancient solution (M~,g 00 (t),p 00 ) to the Ricci flow

in the sense of C^00 Cheeger-Gromov convergence. Then

,lim Vol(gi(O)) = ,lim K;l^2 vol(g(ti))

i--+oo i--+oo

= 00

since Vol (g (ti));::::: c > 0, independent of i.
Now assume M 00 is compact. Then

,lim Vol(gi(O)) = Vol(g 00 (0)) < oo,

i-+oo
which is a contradiction. D

REMARK 17.11 (Noncompact singularity models have infinite volume).
For any finite time noncompact singularity model (M~, g 00 (t)) with bounded
curvature we have Vol (g 00 (t)) = oo. This is because for each t there exists
K, > 0 such that
Vol 900 (t) B 900 (t) (x, 1) 2:: K,

for all x E M 00 (by Perelman's no local collapsing theorem).

Lemma 17.10 implies

COROLLARY 17.12 (Compact singularity model implies>.> 0). Given

a finite time singular solution (Mn, g (t)), t E [O, T), of the Ricci flow, if

some associated singularity model is compact, then>. (g (to)) > 0 for some

to E [O, T); by the >.-monotonicity formula (see Lemma 5.25 in Part I) we

then have).. (g (t)) > 0 for all t E [to, T).