1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

58 28. SPECIAL ANCIENT SOLUTIONS

2 s

By (28.41), we have for the cigar soliton that u(t) = e 4 'e+e 2 s. In particular,

e•/+i :::; u(t) :::; 1 on [O, oo) x 51. With the help of the above lemma, we shall now
prove the following.

LEMMA 28.48 (Lower bound for conformal factor of g(t) relative to 9 cy1). There

exists c( t) > 0, which is positively bounded from below on finite closed intervals, such

that

(28.56) u(s, e , t) 2 c(t) in [1,oo) x 51.

PROOF. Let (ln u )- ~ max{ - ln u, 0} be the negative part of ln u. Then
(ln u)_ 2 0 is subharmonic with resp ect to 9cyl = ds^2 + dB^2 since by (28.52) it
is the maximum of two subharmonic functions. Define the circular averages (times
27r)

(28.57) W _(s, t) ~ { (ln u)_(s, e, t)dB 2 0.

Js,

Since (ln u) is subharmonic, s t--t W(s, t) is weakly convex. Thuss t--t (W_) 8 (s, t)
is nondecreasing and hence the limit

.A(t) ~ lim (W_) 8 (s, t) E [O, oo]

S-400

exists (if .A(t) < 0 , then W_(s, t) < 0 for s sufficiently large, a contradiction).

Claim. For each t we have .A(t) = 0.

Proof of the claim. Suppose that for some t we have .A(t) E (0, oo]. Then there

exists s 0 < oo such that

(W-) 8 (s, t) 2 K,(t) ~min { A;t), 1} E (0, oo) for s 2 so.

Since W(s, t) 2 0, this implies that W(s, t) 2 i<~t) s for s 2 2s 0. Hence we have
exponential decay of the circular minimums:

(28.58)

"(t)

min u(s, e, t):::; e--.:;;-s for each s 2 2so.

8E[0,27r]

This contradicts the estimate in Lemma 28.47 and hence proves the claim.

Now .A(t) = 0 implies thats t--t W_(s, t) is nonincreasing. In particular,

(28.59) W(s, t):::; W(O, t) for s 2 0.

Now let (s, B) E [1, oo) x 51. We have that Bcs,e)(l) is embedded in [O, oo) x 51
and hence is isometric to a unit Euclidean 2-ball. Since (ln u)_ is subharmonic, we
can apply the mean value inequality to obtain that for any ( s, B) E [l, oo) x 51 ,

(lnu)_(s,B,t):::; - -^11 (lnu)_ (s,e, t)dsde

7r Bcs,e)(l)

:::; ! r 1s+

1

(lnu)_(s,e,t)dsde

7r ls^1 s-1

11 s+1

= - W_(s, t)ds

7r s- 1

2

:::;-W_(O,t).

7r