1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

52 28. SPECIAL ANCIENT SOLUTIONS

is continuous, nondecreasing, and surjective. To see the continuity of Gk we argue

as follows. Given any c 0 E [O, oo), since LJ Fk-^1 ([0, c]) n Bk = F;^1 ([0, co)) n Bk

c<co

and by the countable additivity of Riemannian measure, we have

lim_ Gk(c) = Area 9 i(kJ {F;^1 ([0, co)) n Bk}.

c--+c 0

Since L (F;^1 (c 0 )) < oo because of w (Fk,9i(k)) < oo, we have that

Area 9 ,<kJ (Fk-^1 (co)) = 0.

Hence limc--+c- 0 Gk(c) = Gk(co). Similarly, one can show that limc--+c+ 0 Gk(c) =

Gk( co).

In the following, k will be assumed to be sufficiently large. By the properties

of Gk, there exists Ck E ( 0, oo) such that

Area9i(k) (Fk -1 ( [ 0, Ck l ) n Bk) = 7r 2k^2 '

which is approximately half of Area 9 i(k) (Bk)·

Now, since Bk is almost isometric to a Euclidean ball of radius k and since the

set F;^1 (ck)nBk divides Bk into two regions of almost equal areas, it is not difficult

to see that one can deduce from Corollary 28.37 that

Claim.

(28.40)

By (28.40) we have that w (9i(k)) 2 k for all k sufficiently large. This contradicts

the assumption that the widths of (Mi, 9i, Pi) are bounded independent of i.

Proof of the claim. Let B(k) denote the Euclidean 2-ball of radius k. There

exists a diffeomorphism <T>k : Bk ---+ B(k) such that

(1-k-^1 )<T>J;(geuc) :S: 9i(k) :S: (1 + k-^1 )<T>J;(geuc) on Bk·

Then k(F;^1 (ck) n Bk) divides B(k) into two regions of almost equal areas with

respect to 9euc· Hence, by (28.36) we have that

L 9 ,<kJ (F;^1 (ck) n Bk) 2 Vl -k-^1 L 9 .uc(k(Fk-^1 (ck) n Bk)) 2 k. D

Now we consider the example of the cigar soliton. Let ( r , B) be polar coordinates

on IR^2 and define the fl.at cylinder metric 9cyl = d~~!~¥° = ds^2 + dB^2 on IR^2 - {O} 2"

IR x S^1 , where s = ln r. The cigar is the 2-dimensional solution to the Ricci fl.ow

given by (IR^2 , 9cig ( t)), where

(28.41)

. dx2 + dy2 e2s
9cig(t) :::;= e4t + x2 + y2 = e4t + e2s9cyl·

This steady GRS flows along the gradient of the potential function f (x , y) =

• ln ( e^4 t + x^2 + y^2 ) and its scalar curvature satisfies R = ef. At all times t he
  cigar is isometric to

dr^2 + r^2 dB^2

9cig = = dp^2 + tanh^2 p dB^2 ,

1 + r^2

where p ~ arcsinh r = log( r + ~). The geometry of the cigar soliton gives us
some guidance for the proof of the main Theorem 28.41 below.