1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

338 7. THE REDUCED DISTANCE

For example, we may take ( M^2 , g ( r)) to be an evolving 2-sphere and

(N^1 ' h ( T)) to be the line to get the cylinder 82 x JR.

EXERCISE 7.69. Assuming the solution (Mn,g (r)) is defined only for

r > 0, rewrite equation (7.106) using the metric g (1) instead of g (0). Show

that this equivalent form is consistent with (7.108).

We follow up on the above exercise by translating time in our Einstein

solution so that R(O)--+ oo in (7.103), and correspondingly, R(r) --+ ~·

Since we then have d;(O) (p, q) --+ 0, we choose to rewrite the formula for £

in terms of d;(r) (p, q) using

d;(O) (p, q) R (f) 1

2 =--= 2.

dg(r) (p, q) R (0) 1 + ; R (0)

In particular, from (7.105), we have

as R (0) --+ oo.

C(q f)-'!!_ (l-tan-

1

( J27R(O)/n))

' - 2 J2rR(O) /n

R (f) d;(r) (p, q) 1

+ -----,---~------,--=:--;:=:::;=

tan-^1 ( )27 R (0) /n) 2.;2njr R (0)

n

-t -

2

EXERCISE 7.70. Let (Mn, g (t)), t E [O, T), be a maximal shrinking

Einstein solution of Ricci fl.ow so that R (t) = 2 (,P-t)" Given any (xi, ti) E

M x (0, T), we define a solution 9i (r) ~ g (ti - r) of the backward Ricci
flow and lex· ?..' t·) i (x, t) = £^9 (i. Xi, o) (x, ti - t). Check that

+ 4(T-t) ~tan-1 ~·

v '!'=ti v '!'=ti

From the above remarks, if~di) (x, t) --+ ~ as i--+ oo if ti--+ T.

7.2. The£ function on a steady gradient Ricci soliton. Consider

a steady gradient Ricci soliton g( r) = <p;go on Mn where Re (go)+ V'V'fo = 0

and I.PT satisfies

(7.109) 01.fJT or ( ) x = - (grad 90 fo) (I.PT (x)),

<p(O) =idM.