1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

342 7. THE REDUCED DISTANCE

7.3. £function on a gradient shrinker. Let (Nn,h(t)), -oo <

t < 1, be a shrinking gradient Ricci soliton in canonical form as given in

Proposition 1. 7 with E = -1. Define T ~ 1 - t, and let

(7.111) h ( T) ~ h ( 1 - T) and f ( T) ~ j ( 1 - T) ' 0 < T < 00.

Then h ( T) is a solution of the backward Ricci fl.ow on the maximal time

interval (0, oo). Since h(t) = (1-t) 'Pth (0), we have for T > 0

h (r) = Tcp~h (1),

where 'PT ~ 01-T· Although the metric h (0) is not well defined, we may still

define£, L, and£ as before using the basepoint (p, 0). By (1.12) we have

J(r)=f(l)o'PT·

Note that Rh (x, r) = ~Rh(l) ('PT (x)). From

:t 'Pt (x) = 1 ~ t (grad iico/ (0)) ('Pt (x)),

we have

(7.112)

REMARK 7.76. Note that if h (r) is a shrinking gradient soliton fl.owing

along \Jf, then Rc+\7\Jf - 2 ~h = 0. Since f satisfies ~~ = - l'Vfl^2 , the

gradient vector field \7 f satisfies

:T ('VJ)= -2Rc(\7f) + \7 (~~) = -2Rc(\7f)-\7 j\7fj^2 = -~\Jf.

Given a path 'Y: [O, f]--+ N from p to q, its £-length is

£("!)=for Jr ( Rh(T) ("! (r)) +Ii' (r)i~(T)) dr

=for Jr ( ~Rh(l) ('PT (ry ( r))) + T l('PT )* 'Y (r)l~(l)) dr.

Let /3 ( r) ~ 'PT ( 'Y ( r)). Note that the point (/3 ( T) , 1) corresponds geometri-

cally to the point ( 'Y ( T) , r) since in general the point ( x, T) corresponds to

(rpT (x), 1). We have

/3 ( T) = ('PT t ')' ( T) + a:: ('PT ("! ( T))) ,

which implies

. 1
('PT)* 'Y (r) = /3 (r) + - (gradh(l)f (1)) (/3 (r)).
T