1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

156 30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS

(-47rt)-n/^2 e-e;(i.f.>;(x),t)dμi.f.>;g,(t) (x) are bounded above on M 00 by an integrable

n-form independent of i and t. Thus, using the monotonicity of V/,w (t) and the

Lebesgue dominated convergence theorem, we obtain that the limits

(30.92) V 00 (t) = lim Vi (t) = lim Vp,w (w + Ai^1 t) = const E (0, 1]

i---+oo i---too

exist for t E (-oo, 0). By the argument used to obtain Corollary 30.14(3), we

conclude that

1

Re 9 oo(t) + \79oo(t)\79oo(t)f!oo (t) + 2tgoo (t) = O;

that is, g 00 (t) is a gradient shrinker with C^00 potential function /! 00 (t). D

REMARK 30.20. It is possible that g 00 (t) in Proposition 30.19 is flat. For

example, consider the case where n = 3. If there are {pi} and Ai ---t oo such that

_lim Ai^1 1Rm 91 (Pi, w - Ai^1 ) = 0,

t--+00

then the corresponding limit satisfies

= _lim Ai^1 1Rm 91 (Pi, w - Ai^1 )

i--+oo

=0.

Since n = 3, Rm 900 ;:: 0 and, by the strong maximum principle, we conclude

that g 00 (t) is flat fort :::; -1. By Theorem 28.20, g 00 (t) has bounded curvature

on compact intervals in (-oo, 0). Hence, Chen and Zhu's uniqueness theorem for

complete solutions to the Ricci fl.ow with bounded curvature (see [63]) implies that

g 00 (t) is flat for all t E (-oo,O).

We wish to characterize those p E M for which the corresponding shrinker

in Proposition 30.19 is nonflat. Suppose that there exists p E M, ti /' w, and

c > 0 such that (w - ti) 1Rm 91 (p, ti) 2 c for all i. Then any corresponding limit

g 00 (t) is nonfiat (using the rescaling factors Ai~ (w - ti)-^1 ). In this case we have

established the existence of a nonflat gradient shrinker singularity model.

It is not a priori clear whether there always exists such a point p. We know by

the gap theorem (see Lemma 8.7 in [77]) that

1

(w - t) max 1Rm 91 (x, t) 2 -.

xEM 8

This implies that for any ti /' w there exists {pi} such that

(30.93)

Since M is compact and by passing to a subsequence, we may assume that

Pi ---t p for some p E M. However, this does not immediately imply that

(w - ti) 1Rm 91 (p*, ti) 2 c for i large enough and some c > 0. Note that it is

possible that

(30.94). 1. im ( w - ti )-l/2d g(t,) ( Pi,P *)-- oo.

i--+oo