1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

2. BEHAVIOR OF μ (g, T) FOR T SMALL

for all i. Hence, by (17.67), (17.51), and <Pi (0) =Xi,

(17.69)

Wi (0) = Wi (xi) =.mtfXWi 2:: exp(-~ IRmin (g)I - ~ log(27r) - ~).

Hence

(17.70) ti'ioo (0) > 0.

21

By (17.70) and by the strong maximum principle for weak solutions (Lemma

17.26 below) applied to (17.56), we have

W 00 > 0 on ffi.n

and w 00 log w 00 is contained in the local Holder space Cf~~ (ffi.n).

Proof of Step 4. Since w 00 is a weak solution of (17.56), where the RHS

is contained in q~~ (ffi.n), by the regularity theorem for weak solutions we

have that w 00 is a classical solution of (17.56). Now by Schauder theory, we

have for any k E N

. llwoollck,<>(0,gRn) :'.S Ck (0)
for some Ck (0) < oo. In particular, w 00 E W^1 '^2 (ffi.n) n C^00 (ffi.n).

Now we complete the proof of the proposition. For any R > 0 let

'IJR : ffi.n -+ [O, 1] be a radial cutoff function with

(x) = {^1 if^0 :S lxl :S R,

rm O if lxl 2:: R + 1,

and with -2 :S ffr'IJR :S 0. Then by (17.66) with 'P = 'IJAW 00 , we have

[ (Vwoo, v ('IJAwoo)) dμ~,.

}~n

= L,. (μ:; +~log (27r) + ~ + logw 00 ) ('IJRWoo)^2 dμ~n,

so that

0 :'.SL,. l_V ('IJRWoo)l

2

dμ~n

• { (!2'.. log (27r) + !2'.. +log ('IJRWoo)) ('IJRWoo)^2 dμ~n
  }~n 4 2

(17.71) = μ^00 { ('IJRWoo)^2 dμ~n + { (1V'IJRl^2 - 'IJAlog'l}R) w~dμ~,.,

2 }~n }~n

where the inequality is true by (17.46), which holds since f~n w'?x,dμ~n :::; 1.
Taking R sufficiently large, the RHS of (17. 71) is arbitrarily close to

μ00 r ~2 d o

2 }~n Woo μ~n < '

where this inequality is true because μ 00 < 0 and J~,. w'?x,dμ~n > 0. This is

a contradiction. D