1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

166 30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS

Since M 00 is noncompact, by Theorem 9.66 in [77], there exists a sequence of

points { x 00 ,i} in M 00 such that the pointed sequence of solutions

{ (M!,, goo,i (t), Xoo,i)}, t E (-oo, 0],

where

goo,i (t) ~ R 9 = (xoo,i,O)goo (R 9 = (x 00 ,i,0)-

1

t),

converges in the C^00 pointed Cheeger- Gromov sense to an ancient so lut ion of the

Ricci fl.ow (M~, 00 , g 00 , 00 (t), x 00 , 00 ), t E (-oo, O], which splits as the product of IR

with a constant curvature ancient so lut ion on t he 2-sphere. This is our asymptotic

shrinker (recall by Lemma 8.26 in [77] that the limit of a limit is a limit). 0

4.2. Shrinking Ricci solitons are gradient and K-noncollapsed.

In this subsection we show that complete noncompact shrinking Ricci so litons

with bounded curvatures must b e gradient and K-noncoll apsed.

We shall use the following estimate, which is related to (27.19).

LEMMA 30.37 (Bounds for vector fields of shrinking Ricci solitons). L et (Mn, g,

X, -1) be a complete noncompact shrinking Ricci soliton structure with bounded

Ricci curvature, so that

(30.120)

Fi x 0 EM, and let r (x) ~ d 9 (x, 0). Then there exists const < oo such that

(30.121) (X, \7r) (x) ;=:: r ~x) - const

on M - {O}.

PROOF. Let K ~ sup M I Rel < oo. By Proposition 18.8 in Part III, for any

x EM - B (0, 2) and any minimal unit speed geodesic"(: [O, r (x)] ---+ M joining

0 to x, we have

for(x) Re("!' (s) ,"(^1 (s)) ds::::; 2 (n - 1) (~K + 1).

Using (30.120), we obt ai n

r (x)

(X,\7r)(x)- (X(0),1'(0))= Jo (\7'Y'(s)X,1

1

(s))ds

r (x) ( 1)

= J

0

- Re ("!' ( s) , "(^1 ( s)) +

2

ds

;::: ~r (x) - 2 (n - 1) (~K + 1).

0

Partially extending Corollary 27.7, we have

LEMMA 30 .38 (Nongradient shrinking Ricci solitons and completeness of their

vector fields). If (Mn, g, X, -1) is a complete noncompact shrinking Ricci soliton

structure with bounded Ricci curvature, then the vector field X is complete.