1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

300 7. THE REDUCED DISTANCE

SOLUTION TO EXERCISE 7.21. Consider a maximal Einstein solution

to the backward Ricci flow h (r), TE [O, T), where Re= - 2 (i-T)h is nega-

tive. In this case h (r) = TTT h (0) (which is easy to see from Rch(T) being

independent of r). The £-geodesic equation (7.32) for/ (r) is

(7.38) V'xX +(I_ --

1

-) X = 0.

2T T-T

Let f3 (p) ~ / (J (p)) , where 1 is defined by

J' u-l (r)) ~exp (/

7

( 2 ~ - T ~ r) dr)

=fi(T-r),

i.e., l' (p) = VTfP) (T - 1 (p)). Then /3 (p) ~ ~ = ~; (J (p)) l' (p). Let

T ~ 1 (p). Equation (7.38) and the definition of 1 imply

\i' /3(p)/3 (p) = \i' f'(p)!r;(T) (t (p) ~~ (7))

= J" (p) d1 (r) _ l' (p) l' (p) (I_ __ 1_) d1 (r)

dr 2r T-r dr

= !!__ (1' (p) - exp (Jf(p) (

1

_ - ~) dr)). di (r)

~ ~ T~r ~

=0.

That is, f3 (p) is a constant speed geodesic with respect to h (0). We make

the rationalizing substitution x : VTfP), so that

=! l'(p)dp = 2 / dx =-l lo (VT+x)

p VTfP) (T - 1 (p)) T - x^2 VT g VT - x ·

That is,

1 (p) = x2 = T (-e~_TP-_1)2

evTp + 1

or

1-1 (r) = _1 log (VT+ VT).

VT VT-./T

Using I ( T) = f3 u-^1 ( T)) and the fact that f3 is constant speed with re~pect
to h (0), we compute

l

dr/

2

= T-r j/J(J-l(r))j2 d(J-1)

2

dr h(T) T h(O) dr Tr (T-r)'

canst

since h (r) = TY,^7 h (0) and

d u-1) - (^1 1)

dr l'(p)