1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

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

Sesum [169] considers immortal solutions g (t) to the 'Ricci flow with

cosmological. constant l'

(17.38)

with

(17.39)

8

-g = -2Rc+g

at

IRml :::; C and diam :::; C

on M x [O, oo), where C < oo. Making the change of time variable t (i) ~

- ln (1 - i), i.e., i (t) ~ 1-e-t, and rescaling the solution by defining g (i) ~

( 1 - i) g ( t (£)) , we have

-g^8 ~ = -2Rc ---

ai

on M x [O, 1). The conditions (17.39) correspond to the Type I condition

I Rm I ( i) :::; --2___,_ 1-t

and the diameter estimate

(17.40)

This implies

(17.41)

diam ----:::; Cv ~ 1 - t.

For if the (Ricci) curvature were uniformly bounded, then the diameter could

not tend to zero as i --+ 1.^2 Moreover, we then also have

max !Rini (i) > (

1

M -8 1-t A)

(see Lemma 8.19 in Volume One and Lemma 8.7 in [45]).

In turn, if one does a Type I rescaling by defining

9i (t) = 1 ~ -t/ (ii+ (1 - ii) t)

for some ii--+ 1, then one obtains uniform bounds for IRmil (t) and ~i (t)

for t:::; 0.

PROBLEM 17.17. Show that if (Mn,g(t)), t E [0,T), is a finite time
singular solution forming a singularity model on a closed manifold (which
then must be diffeomorphic to M), then g (t) is Type I.

(^2) This implies
m,tt Jifr.;!J (ti) ---+ oo for some ti ---+ 1,
from which (17.41) follows.