1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

3. FIRST VARIATION OF £-LENGTH AND EXISTENCE OF £-GEODESICS 299

EXAMPLE 7.20 (£-geodesics on Einstein solutions). Let (N 0 , ho (T)),

T E (0, oo) , be a 'big bang' Einstein solution to the backward Ricci flow

with Reho (T) h~~). Then Rho (T) = ~ and the £-geodesic equation

(7.32) is

(7.36)

3

'\lxX +-x = o,

2T

so that '! iki. (7^3 /^2 ~) = 0. Note that since '! is independent of scaling and
dr.
ho (T) =Tho (1), we have '\lho(^7 ) = '\lho(l) is independent of T. Clearly the
constant paths, where X = 0, are £-geodesics. More generally, reparametrize
'Y and define the path /3 by /3 (p) = "( (f (p)) , where

(7.37) T3/2 = f' u-1 (T)).

Then ~(p) ~ = ~(f(p))f'(p) = ~;(f(p))f(p)^312 , so that (7.36)
implies

'\J /j(p)~ (p) = '\J 7 3/2~(T) ( T^3 /^2 ~; ( T)) = 0 i

i.e., /3 is a constant speed geodesic with respect to ho (1). Since solutions of
(7.37) are given by f (p) = -( po-p^4 ) 2 , the £-geodesics are of the form

"( ( T) = j3 (~ - 3) '

Fa VT

defined for T E (0, oo), where /3 : (-oo, oo) ----+ No is a constant speed
geodesic with respect to ho (1). Note that

I

d"( I

1

-(T) =VT /3. ( ---^2 2 ) --^1 I

dT ho(T) VTo VT^7312 ho(l)

canst

T

That is,

T^2 I ~'Y (T)\

2

=canst.

T ho(T)

(Compare with (7.26) for the Ricci fiat case.) In particular, I~; (T)I =

ho(l)

c_.; 3 %t. In any case, the speed of ~; ( T) tends to infinity as T ----+ 0, whereas

the speed of¥ T (T) tends to zero as T----+ 00. Note rXl TO 1~; (T)I ho(l) dT < 00

for all To E (0, oo), whereas J;^0 I~; ( T) I dT = oo.

ho(l)

EXERCISE 7.21. Determine the £-geodesics for Einstein solutions with

negative scalar curvature. What multiple of I ~'Y ( T) 1

2

T h(T) is constant?