1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

344 7. THE REDUCED DISTANCE

Following up on details related to the previous exercise, we have

EXERCISE 7.80. In regards to (7.114), show that although for <.p 7 (q) # p

a smooth minimizer of inf,e J; T^312 I~ (7)1

2

dT, where the infimum is taken

h(l)

over paths joining p to <.p 7 (q), does not exist (which i!3 related to h (0) not

being well-defined), any minimizing sequence f3i of paths joining p to <.p:r-(q)

limits to the constant path T 1--+ <.p:r-(q). That is, any minimizing sequence

'Yi of paths joining p to q for 2 ~£ ("!)limits to the path T 1--+ <.p-;^1 (<.p:r-(q)).

Note that in general, q 0 ~ limr-->O <.p-;^1 (<.p:r-(q)) # p. We may think of this

as saying that the minimal geodesic starting at (p, 0) immediately jumps to

(qo, 0) and then becomes a constant path in the geometric sense. Caveat:

The solution is undefined at the 'big bang' time T = 0.

We now present another proof of Lemma 7.77, following an original idea

of one of the authors [289]. (The proof given above is also inspired by his

line of reasoning.) Given a path "! : [O, r] --+ N, we have

d~ (.Jif("!(T),T)) =.Ji(fr + ~~ +\7f·'Y)

(7.116) =.Ji (L +

81

+~1v11^2 + ~ l'Yl^2 - ~ l'Y-v11^2 ).

2T OT 2 2 2

Now for a gradient shrinker in canonical form, where (1.14) holds, i.e.,

(7.117) ~~ = - IV Jl

2

,

assuming f has a critical point,^12 we may normalize f (by adding the ap-

propriate constant) so that

. 1
(7.118) R+ IVJJ^2 - -f T = 0.

Substituting this with (7.117) into (7.116), we have

d~ ( .Jif ("! (^7 ) ,T)) =~VT ( R + J'YJ

2

• J'Y-VJJ

2

).

Hence

f ("! (r), r) =

2

~£ ("!) -

2

~ 1

7

.Ji l'Y (T) - V f ("! (T), T)i~(r) dT.

Given a point q E N, we may take "( : (0, r] --+ N to be the path with

'Y (T) = V f (T) for all TE (0, r] and"! (r) = q. We have

1

f ("! (r), r) =

2

y!T.c ("!) ;::: .e ("! (r), f).

Since T = 0 is the big bang time,£("! (r) 'r) is independent of the basepoint
chosen.

(^12) If N is compact, this assumption is always satisfied (though it is also satisfied for
the shrinking Gaussian soliton).