1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

102 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

9.2. A concentration-compactness result.

The following is a concentration-compactness-type or bubbling-type result. In

view of the geometric perspective, note that the area form of t he pull-back g(t)

of g(t) to IR^2 is u(t)dμeuc and a lso that the Cohn-Vossen inequ ality says t hat for

a complete noncompact Riemannian surface (M^2 , g) with positive curvature, we

have JM R 9 dμ 9 ~ 47r. Throughout t his subsection, all balls are with respect to t h e

Euclidean metric on JR^2.

LEMMA 29.33 (u unbounded implies a concentration of integral of curvature).

Suppose that u 00 (0) < oo and let z 0 E IR^2 , r > 0, and c > 0. Then there exists a

constant C = C(lzol, r,c,u 00 (0),umin(-l)) such that if

( Rg(t)il(t)dμeuc ~ 47r - c

} Bz 0 (r)

at some t E (-oo, -1], th en at this time t we have

u(t) ~ e^0 in Bz 0 (r/2).

REMARK 29.34. In the case of a round 2-sphere shrinking to a point at time 0,

we h ave by (29.6) that

• 8 ltl
  u(r,B,t)=( 2 ) 2.

l+r

In this case, u 00 (r, B) = oo. Moreover,

1 1

r l67r 87rr^2

Rg(t)il(t)dμeuc = ( 2 ) 2 rdr = -- 2 ·

Bo(r) o 1 + r 1 + r

So we cannot simply remove the condition u 00 (0) < oo from t he hypothesis of the

lemma.

PROOF OF LEMMA 29.33. Let w be the unique solution to

-.6.eucW = R9u in Bz 0 (r),

w = 0 on 8Bz 0 (r).

By Lemma 28.50 we have that for each o E (0 , 47r),

(29. 10 3)

1 (

o lw(x)I ) d l67r

2

r

2

exp μ euc < ---.

Bz 0 (r) llR9ullL1(Bz 0 (r)) - 47r - 0

Since I IR9ul lu (Bza (r)) ~ 47r - c from the hypothesis, by choosing o = 47r - ~, we

h ave

(29.104)

1

qlw(x)ld 327r2r2

e μ euc ~

Bz 0 (r) €

for some q;:::: 88 ;~ 2 ° 0 > l. By Jensen's inequ ality we h ave

(29.105)

1

I w - id μ euc ~ 7rT2 - 1 ( n -^1 2 1 e qlwl dμeuc )

Bz 0 (r) q 1rT Bza (r)

~ 7rr^2 87r- 2c 1 ( 327r). C( )

8

n - =;= r , c.

7r - € €