1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

4. ANCIENT SOLUTIONS WITH POSITIVE CURVATURE 65

By Lemma 6.35(2) in Part I (note that,\ (g (t)) > 0 since R (g (t)) > 0), we conclude

that g (t) is a gradient shrinker and, by (2), that g (t) must have constant sectional

curvature. 0

PROBLEM 28.57. Can one remove the n;-noncollapsed condition in Theorem

28 .56?

4.2. Ancient 3 -dimensional solutions with pinched Ricci curvatures.

In the presence of positive curvature, instead of assuming the T yp e I condition

as in the previous subsection, we may assume a curvature pinching condition. The

following result is due to Brendle, Huisken , and Sinestrari [34] (they also prove a

higher-dimensional analogue). This result is related to Hamilton's "necklike points

theorem" (see Theorem 9.19 in Volume One).

THEOREM 28.58 (Compact 3-dimensional ancient solutions with pinched Ricci

curvatures). If (M^3 ,g(t)) , t E (-oo,O], is an ancient solution of the Ricci flow

on a closed 3-manifold with Re ;::: cRg, where c > 0 is a constant and the scalar

curvature R is positive, then g (t) has constant positive sectional curvature.

PROOF. Let ,\ ;::: μ ;::: v denote the eigenvalues of Rm (equal to twice the

0

principal sectional curvatures) and let Rm denote the trace-free part of Rm. Given

E: > 0, define

(^0 2 2 2)

G = 1Rml2 = (μ - v) + (,\ - v) + (,\ - μ) > 0

(28.74). R 2 - € 3R 2 -€ -.

By the formula on p. 273 of Volume One with / = 0, we obtain

(28.75)

a 2(1-c)

at G:::; ~G + R (\JG, \l R) + 2J,

where

(28. 76)^1 (

0

J ~ R3-€ clRml2 IRml^2 - p )

and

(28.77)

On the other hand, the hypothesis Re ;::: cRg yields

μ+v

μ;:::-

2

• ;:::cR;:::clRml.

Thus

(28.78) P;::: μ^2 (,\ - v)^2 ;::: c^2 IRml^2 1Rml°^2 ,

2 0

using(,\ - v) ;::: 1Rml^2.

2

Now take E: = T· Then, by (28.78), we have

(28. 79)

0 2

J < -E: 1Rml

2

IRml < -E:Gl+~

• R^2 -€ R - '

('R~l

2

r (which is equivalent to 11*11 :::; ('~^1 ) \ There-

0?
W h ere. we use d IRml-R2-e :::;

fore

(28.80) .!!_G < ~G +

(^2) (l -c) (\JG \l R) - 2cGi+~.
at - R '