1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

2. BEHAVIOR OF μ (g, T) FOR T SMALL 19

Since gi = if!igi---+ 9JR.n, by (17.116) below and (17.60), we have for any

compact domain n C ~n and any 2 < p < n^2 ::_ 2 (when n = 2, define

n-2 2n ::'::: · oo)

(17.61)

where C (O,p) < oo is independent of i. Since (17.53) says

(17.62)

-2Ll 9 iwi + ~R 9 iwi - (~log (27r) + n) Wi - μ (gi, ~) Wi = 2wi logwi,

by the LP estimate for solutions to second-order elliptic equations (see The-

orem 9.11 in Gilbarg and Trudinger [71]) applied to wi on n, we have

(17.63) llwi llw2,p(n, 9 1&\n) :S C (n, p)

for any 2 < p < n^2 ::_ 2 and all i.

By the Sobolev inequality, we have

1

C llwill L P( n=-2 n )2 (!1,91&\n) :S llwillW1'~(!1 ,gl&\n ) :SC llwillw2,p(!1,gl&ln),

where C = C (O,p) < oo is independent of i (note that n":p 2: n":. 2 and

JM w'f dμ 9 i = 1). Thus, by applying (17.115) with 8 > 0 arbitrarily small,

we have

lo lwdogwilq dμJR.n :SC (n, q)

for 2 :S q < 2 ( n":._ 2 )

3

, independent of i. From this and the standard elliptic

LP estimate for (17.62), we obtain the stronger (as compared to (17.63))

estimate

llwillw2,q(n, 9 l&ln) :SC (n, q)

for 2 :S q < 2 ( n":._ 2 )

3

. By iterating the above argument (easy exercise), we

see that for any q E (1, oo)

(17.64) llwillw2,q(n, 9 l&ln) :SC (n, q),

where C (n, q) is independent of i.^5

Because we have (17.64) with q > n, by the Sobolev inequality it follows

that

llwillc1,a(n, 9 l&ln) :SC (0)

for some a E (0, 1) and where C (0) < oo is independent of i. By the Arzela-

Ascoli theorem and a diagonalization argument, passing to a subsequence,
we have that for some a E (0, 1) there is a nonnegative function Woo in
Cl,a (~n, 9JR.n) such that