1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

1. REDUCED DISTANCE OF TYPE A SOLUTIONS 141

PROOF. Let (q, l) E JC x [a 00 + c:, w 00 - c:]. By Lemma 7.34 in Part I, the time

derivative of Li is given by

(30.28)

By the Type A assumption and by Lemma 30 .7 , we have

1

-Jwi - lRg, (<Pi(q), l) + Jwi - lh~lg2, (t)I :=; Cn ' + cons_t '.

(wi - l)"-2 (wi - t )2

The lemma easily follows. D

EXERCISE 30 .9. Derive explicit bounds for !Viii and J~J, in terms of time

and t he distance to a fixed point, as in Lemma 30.4.

1.5. The limit reduced distance and its properties.

Now that we have estimated the L-distances for the sequence of solutions gi (t),

we shall investigate some of the properties of the limit of the reduced distance

functions.

THEOREM 30.10 (Reduced distance based at the singular time for a sequence of

solut ions). Suppose that complete solutions with bounded curvature (Mf,gi (t) , Pi),

t E [ai , wi], converge in the C^00 pointed Cheeger- Gromov sense to (M~, g 00 (t), p 00 ),

t E [aoo, w 00 ) , where Wi / w 00 and ai :=; a 00. Assume the Type A condition that

M

IRmg,I ::; ( Wi - t r on Mi x ( O'.oo' Wi)

for some M < oo and r E [1, ~) independent of i. Then there exists a subsequence

such thatli ( q, l) = e~; ,w, (<I> i ( q), l), where the <I> i are as in ( 30. 6), converges in C 1 ~c

for a E (0, 1) to a function

(30.29)

which is called a reduced distance based at the singular time and which sat-

isfies the following properties:

(i) R 00 is locally Lipschitz.

(ii) R 00 satisfies

(30.30)

fJe 00 2 n

-fit -6.goofoo + l'Vfoolgoo - Rg= + 2 (woo - t) 2': 0

and holds both in the support sense and in the weak (distributional) sense,

respectively.

A direct consequence is the following.

COROLLARY 30.11 (Reduced distance based at the singular time for a Type A
solution). Let (Mn,g(t)), t E [O,T), be a complete Type A singular solution with

T < oo. For any p E M and Ti / T , there exists a subsequence such that the

reduced distances fp,T ; converge in C 1 ~c for a E (0, 1) to a reduced distance based at

the singular time, i.e.,

(30.31) fp,T : M x [O, T) --t IR,