1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

30 17. ENTROPY, μ-INVARIANT, AND FINITE TIME SINGULARITIES

for s E (0, a]. If t E (0, a], then

F(t) ~ C (lot brdr+ (~ +b) lot r~=l for sn-ids)

=C(b+~:b)~

since -x log x ~ ~. Choosing a small enough, we have

F (t) ~ t

fort E (0, a].

In general, if we have F (t) ~ tk on (0, a] for some k 2 1, then

F (t) ~ C (lot rk+^1 dr - lot r~=l for sn-l+kk logs ds)

+ C -- sn-l+kds.

i

t dr ir

o rn-1 o

Applying -x log x ~ ~ again and integrating, we obtain

(

F (t) ~ C -k-tk+2 + -k it d : ir it k+l )

1 sn-

(^2) +kds + !___kdr

• 2 e o rn o o n +
  (17.87) (
  tk+^2 k tk+l tk+^2 )

= C k + 2+;(k+1) (n + k -1) + (k + 2) (n + k)

provided t E (O,min{a, 1/e}], where C is independent of k. Now (17.87)
implies that there exists ao E ( 0, min {a, 1 / e}) independent of k 2 1 such
that^11
F (t) ~ tk+I/^2
fort E (0, ao]. By induction, we have

F(t)~tf-
fort E (0, ao] and all f 2 1. This implies F (t) = 0 fort E (0, ao] and Lemma
17.26 is proved. D

3.3. The maximum value of a minimizer.

Under the constraint JM w^2 dμg = 1, let ilh be a minimizer of the func-

tional/(, (g, w, r) defined by (17.7). The maximum value of w 7 is related to

an upper bound for the μ-invariant as follows (this is used in the proof of
Proposition 17.20).

(^11) Indeed, we just need a small enough so that

(

t2 k t t^2 )

C k+2+~(k+1)(n+k-1)+(k+2)(n+k) S.Vt

for t E (0, a] and k ;:::: 1.