1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

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

Without loss of generality, we may assume r = 1. Recall from (17.8)

and (17.7) that

μ (g, 1) =inf { }{ (g, w): w E W^1 '^2 (M, g), JM w^2 dμ = 1},

where

1-L (g, w) ~ JM ( 4 l\7wl^2 + ( R - log ( w^2 ) - ~log ( 47r) - n) w^2 ) dμ

for w E W^1 ,^2 (M,g). By Lemma 17.5, the functional 1-L (g, ·) is bounded
from below.

STEP 1. There exists a minimizer 0 s Woo E W^1 >^2 of 1-L. Let { WihEN
be a minimizing sequence of W^1 >^2 functions for the functional 1-L (g, ·) with
JM wrdμ = 1 for all i E N. We may assume that Wi 2: 0 for the following
reason. If w E W^1 ,^2 , then lwl E W^1 ,^2 and

IV' lwll S IVwl

(see Corollary 2.1.8 of Ziemer [198]), so that 1-L (g, lwl) s 1-L (g, w). Thus, if

{wi} is a minimizing sequence in W^1 ,^2 , then so is {lwil}.

Recall that we proved that there exists C < oo (independent of i) such

that

(17.73)

for all i E N (we leave this as an exercise; see p. 238 in Part I or (17.58)
above). By the Banach-Alaoglu theorem,^8 there exists

Woo E W^1 '^2 (M, g)

and a subsequence such that Wi converges to w 00 weakly in W^1 ,^2 ( M, g), i.e.,

for every v E W^1 •^2 (M,g)

i~~ (wi, v)w1,2(M,g) = (woo, v)w1,2(M,g).

As a standard consequence, we have (see also p. 205 in Part I)

(17.74) /lwoo/lw1,2cM,g) s l~~f llwillw1,2cM,g).

By (17. 73) and the Rellich-Kondrachov compactness theorem, for every

c: E (0, ~~~]we have that Wi converges to w 00 in L;;:::.2-ro (M,g). In particu-

lar, JM w~dμ = 1 and w 00 2: 0. By this convergence and by Lemma 17.25
below, we have

μ (g, 1) S 1-L (g, w 00 ) S .lim 1-L (g, Wi) = μ (g, 1).

i-too

We conclude that

(17.75) }{ (g, Woo) = μ (g, 1).

(^8) See Theorems 3.15 and 3.17 in Rudin [164].