1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

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is

7. THE REDUCED DISTANCE

d;(O) (p, q)

1; </> (T)-l dT

and the minimizer is given by a minimal geodesic 'Y with speed

I

d"( I. 1 dg(O) (p, q)

dT g(O) = </> (T) 1; </> (T)-l dT.

Caveat: Here we have assumed that the improper integral 1; </> (T)-^1 dT is

convergent.

Hence

inf r VT (1 +

27

R (o)) I dd"( 1

2

dT

} 0 n T g(O)

d;(O) (p, q)

1; T-^112 (1+^2 : R(o)r^1 dT.

Again we make the rationalizing substitution O' = 2y'T to get

-^1 /^2 1 TR 0 d - n d

1

'f ( 2 )-l 2 1

2

..ft 1

0

(^7) +-:;;- ( ) (^7) - R (0) o 2nR (0)- (^1) + 0' (^2) (]'

• 
  
  
  
  • v2nR ~ ( )-1/2 0 tan -1 ( 2y!T )
  
  
  
  

ffnR(O)-^1 ;^2 ·

We conclude

LEMMA 7.67. Let (Mn, g (T)) be an Einstein solution of the backward

Ricci flow on a time interval containing 0 with R (0) > 0. Let p EM be the

basepoint. For all f E (0, oo) , we have

In particular,

(7.105)

L (q f) = nvfr (1-tan-1 ( ..j2f R (0) /n))

' ~-Jr2=f=R=(o=)=/n=-~

. R (0)1/2 d;(O) (p, q)

• ffn tan-^1 ( ..j2f R (0) /n).

_ n·( tan-

1

(..j2fR(O)/n))

£(q,T)=2 l- ..j2TR(O)jn

..j2f R (0) /n d;(O) (p, q)

+ tan-^1 ( ..j2f R (0) /n) 4f.