1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

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

STEP 1. For a subsequence, the functions

wi ~ wi o <I> i : ui --+ JR

converge in C^1 ,a on compact subsets to a C^1 ,a function w 00 on JR.n, for some

a E (0, 1). Moreover, w 00 E W^1 '^2 (JRn) and

(17.55)

STEP 2. The limit function w 00 is a weak solution to the elliptic equation

(17.56) 2~~nW 00 = - (μ 00 +%log (27r) + n + 2logw 00 ) W 00 •

STEP 3. The function Woo is positive (by Step 1, this implies the cl,a
convergence of wt log Wi to w~ log Woo).
STEP 4. The function w 00 is C^00 and from the elliptic equation (17.56)
that w 00 satisfies, we obtain a contradiction to (17.45) since (17.55) holds.
Now we prove Steps 1-4. From (17.50) and (17.7) we have for the min-
imizers Wi of K, (gi, ·, ~) that

μ (gi, t) = '/(, (gi,Wi, ~)

(17.57) = { ( ~ ( R^9 iwt +^4 JV'wiJ;i) ) dμg··

JM - (log (wt) + ~ log(27r) + n) wt '

Proof of Step 1. By (17.57) and (17.54) we have

(17.58)

2 JM JV'wiJ;i dμ 9 i = 2 JM wt logwidμ 9 i - JM tR 9 iwtdμ 9 i

• μ (gi, t) + % log(27r) + n

:S C1,

where C1 is independent of i; here we used the logarithmic Sobolev inequal-
ity, Lemma 17.2, and R 9 i = 2TiR 9. Hence there exists C 2 < oo such that

(17.59)

for all i. By the L^2 Sobolev inequality, i.e., (17.18), we then have^4

(17.60) llwiJIL~(M,gi) :S C3

when n > 3 and we have JlwillLP(M,gi) ::::; C4 (p) for all p E [1, oo) when
n= 2.
Now consider

(^4) From (17.19), the L (^2) Sobolev constant is independent of scaling and Vol (gi)- (^2) /n :::;
Vol (g)-^2 /n.