1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

7. THE £-FUNCTION ON EINSTEIN SOLUTIONS AND RICCI SOLITONS 343

Hence

;:, ('Y) = r Vi(.!. Rh(l) (/3 ( r)) + r [/3 (r) + .!. (grad h(l)f (1)) (/3 ( r)) [

2

) dr

lo r r h(l)

= 1

7

r-^1!^2 ( Rh(l) (/3 (r)) +I (gradh(l)f (1)) (/3 (r))l~(l)) dr

• r r^312 l/3 (r)l

2

dr + r 2./idd (f (/3(r)'1)) dr

lo h(l) lo r

= 1

7

r-^1!^2 (Rh(l) (/3 (r)) +I (gradh(1)f) (/3 (r), l)l~(l) - f (/3 (r), l))dr

+ f

7

r^312 l/3(r)l

2

dr+2~f(/3(f),1).

lo h(l)

On a shrinking gradient Ricci soliton we have

(7.113) Rh(l) (/3 (r)) +I (gradh(l)f) (/3 (r), l)l~(l) - f (/3(r),1) = 6.

Hence

and

1E£('Y)=J(/3(f),1)+6+ lE {7r3/2l/3(r)l2 dr.

2vr 2vr lo hc1J

We have

f (/3(f)'1) = f (<,0~^1 (/3 (f)) 'f) = f ('Y (f) 'f).

Note that from (7.107),

(7.114) inf {

7

r^312 l/3 (r)l

2

dr = 0,

/3 lo h(l)

where the infimum is taken over all f3 : [O, f] --+ N joining p to <p 7 ( q). Since

f3 (f) = <p 7 (q) implies 'Y (f) = q, we conclude

LEMMA 7. 77 (Reduced distance on shrinker). For a shrinking gradient

Ricci soliton as in Proposition 1. 7 with c = -1,

(7.115) /!, (q, f) = f (q, f) + 6,

where f is defined in (7.111) and 6 is from (7.113). That is,

/!, (q, f) = f (/3(f)'1) = f (<,07(q)'1) + 6,

where f3 ( r) ~ <p 7 ('Y ( r)) and <p 7 is defined by (7.112).

REMARK 7.78. Note that for flat Euclidean space, thought of as a shrink-

ing (Gaussian) soliton, the potential f = I~~ given in (1.16) is the same as
e.

EXERCISE 7.79. Show that for a shrinking gradient Ricci soliton, the
paths 'Y (r) = <,01- 7 (x) = cp 7 (x) for x EN fixed are £-geodesics.