1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

60 28. SPECIAL ANCIENT SOLUTIONS

We shall use the following Trudinger- Moser-type inequality of Brezis and Merle

[36].

LEMMA 28.50. Suppose that B(r) C IR^2 is a ball of radius r and that w : B(r) ---+

IR is a C^2 ,a. function satisfying w = 0 on 8B(r). Then for each 6 E (0 ,4n) we have

the estimate

(28.61)

1 (

6 lw(x)I ) d ( ) 16n

2

r

2

exp μ"fE x < --.

B(r) 11.0.wllu(B(r)) - 4n - 6

PROOF. Define ii; : IR^2 ---+ IR by

w(x) = J_ r ln (-1

2

r 1) l.0.w(y)ldμ"fE(y).

2n j B(r) X - Y

Since - 2 ~ ln Ix - YI is the fundamental solution to the Laplace equation on IR^2 , we

have (see Theorem 4.3 of [122] for example)

.0.w = -1.0.wl in B(r).

If x E B(r), then w(x) :'.:'. 0 since lx~yl :'.:'. 1 for y E B(r). Since .0.w ~ ±.0.w in B(r),

ii; :'.:'. 0 on 8B(r), and ±w = 0 on 8B(r), by the maximum principle we conclude

that w :'.:'. ±w; that is, w :'.:'. lwl in B(r).

Recall t hat by J ensen's inequality we have for functions cp and 'ljJ ;:::: 0, with

JB(r) 'ljJdμ"fE = 1, that

exp ( r cp'ljJdμ"fE) ~ r e'P 'ljJdμ"fE.

}B(r) }B(r)

T ka mg. <p ( ) y _ - - 2 ii 7l"^1 n ( - 1 X ._ 2r Y 1 ) an d · '·( 'I-' y ) _ - 11 <..>.W A l~ 11 w(Ll y)I (B(r)) , we o abt m ·

=exp ( ( _!_ ln (~) l.0.w(y)I dμE(Y))

JB(r) 2n Ix -yl ll.0.wllu(B(r))

ow(x)

/;

< ( (~),;;: l.0.w(y)I dμ"fE(y).

• JB(r) Ix -yl ll.0.wllL'(B(r))

Using w :'.:'. lwl and integrating the above inequality yields

/;

< ( ( (~),;;: dμ"fE(x ) l.0.w(y)I dμ"fE(y).

• JB(r) JB(r) Ix -yl 11.0.wllL'(B(r))
  On the other hand, since x , y E B(r),

1 (

2r )

20

-I -I ~ dμ"fE(x) ~^1 ( -I 2r I ) /~ dμ"fE(x )

B(r) X - Y B(r) X