1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

406 8. APPLICATIONS OF THE REDUCED DISTANCE

Therefore

1 2 n-2 n-2

Noting that rn-^2 e-2r :::;; (n - 2)2 e--2-, we get

V2 - ( Er 2) :::;; Wn-I (n - 2)n-2^2 e--n-2^2 - exp ( - 1 )

2

.J€.

D

4. Backward limit of ancient K-solution is a shrinker

DEFINITION 8.31 (ancient 11;-solution). Let /1; be a positive constant. A
complete ancient solution (Mn, g(t)), t E (-oo, OJ, of the Ricci fl.ow is called
an ancient 11;-solution (or 11;-solution for short) if it satisfies the following
three conditions.

(i) g(t) is nonfl.at and has nonnegative curvature operator for each

t E (-oo, OJ.

(ii) There is a constant C < oo such that R9 (x, t) :::;; C for all (x, t) E

M x (-oo, OJ.

(iii) g(t) is 11;-noncollapsed on all scales for all t E (-oo,OJ; i.e., for any

p > 0 and for any (p, t) E M x (-oo, OJ, if !Rm 9 (x, t) I :::;; p-^2 for

all x E B-g(t) (p, p) , then

Vol-g(t) -~~~--->;;,. B-g(t) (p, p)

pn -

If the curvature bound condition (ii) in the definition is replaced by the

requirement that g ( t) satisfies the trace Harnack inequality

8R

at+ 2\i'R · X + 2Rc(X,X) 2:: 0 for all X,

we say that g (t) is a 11;-solution with Harnack. In Part II of this volume
we will prove that in dimension n = 3 the notions of 11;-solution with Har-
nack and ancient 11;-solution are equivalent. In this section we prove that
in all dimensions certain backward limits of ancient ;;,-solutions are nonftat
shrinking gradient Ricci solitons (Theorem 8.32 below). The proof will take
several steps, which will be carried out in the following subsections.

4.1. Statement of the theorem. Let (Mn,g(t)), t E (-oo,OJ, be a

K-solution. For to E ( -oo, OJ we define a solution to the backward Ricci fl.ow

(M,g(r)), TE [O,oo), by

g (r) ~ g (to - r).