1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

104 29. COMPACT 2 -DIMENSIONAL ANCIENT SOLUTIONS

9.4. Proof of the classification of v 00 •

We now classify the backward pointwise limit v 00. Let 'ljJ E [-~,~]denote the

latitude on 52.

PROPOSITION 29.36 (Infinite expansion or cylinder limit). Assume that the

ancient solution ( 52 , g( t)) is not round. Then, after pulling back by a suitable

conformal diffeomorphism <P of 52 , we have the following. The v(t) converge in

C^1 '°' on 52 and in C^00 on compact subsets of 52 - { N , S} to

(29.107)

for someμ> o. In particular, Voo E C^00 (5^2 ) and Roo = 0 on 52 - {N, S}.

PROOF. By Proposition 29.16, Lemma 29.35, and applying a conformal diffeo-

morphism, we may assume that v 00 is nonzero on n ~ 52 - { N, S}. With this, we

h ave that v 00 ~ limt-+-oo v(t) > 0 on all of IR x 51 , where vis defined in (29.4).

Recall from §6 of this chapter that Voo E C^00 (D). From Lemma 29.12(2), we

h ave that v (t) converges to Voo in C^3 '°' on compact subsets of n. By this and

(29.52), we conclude that R 00 = 0 in D. Hence .6cyt ln v 00 = 0 on IR x 51.

By (29.5) and v 00 :::; C , we h ave that

(29.108) ln v 00 (s, e) = ln(cosh^2 svoo('l/J, B)):::; C1lsl + C2,

where C 1 and C 2 are nonnegative constants. Now pull b ack ln v 00 by the covering

map 7r : IR x IR-+ IR x 51 to a harmonic function w 00 on IR^2. Then

(29.109)

Claim 1. There exist nonnegative constants C 3 and C 4 such that

(29.110)

Proof of Claim 1. By (29.109), we only need to prove a lower bound for w 00. By

applying the mean value property and then using - lw 00 I 2 w 00 - 2(C1lsl + C2)

from (29.109), we have that for any r > 0 and z 1 E Br(O),

(29.111) 7rr^2 (woo(zi) - woo(O)) = { woo(z)dμeuc - { w 00 (z)dμeuc

} B , 1 (r) } Bo(r)

2 - f lwoo(z)I dμeuc

J B, 1 (r)6Bo(r)

2 { (woo(z) - 2 (C1lsl + C2))dμeuc·

J B, 1 (r)6Bo(r)

Note that the symmetric difference satisfies B z, (r).6Bo(r) C B 0 (r + lz 1 I) -

Bo(r - lz1I). Now (29.111) yields

Kr^2 (woo(zi) - w 00 (0)) 2 { woo(z)dμeuc

} Bo(r+lzil)-Bo(r-lz11)

• 2 { (C1lsl + C2) dμeuc·
  } Bo(r+lzil)-Bo(r-lz1 I)

Applying the mean value theorem to the first t erm on the RHS, we obtain

7rr^2 (woo(z1) - woo(O)) 2 47rrlz1l(woo(O) - 2C1 (r + lz11) - 2C2).