1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

3. 2-DIMENSIONAL ANCIENT SOLUTIONS WITH FINITE WIDTH 53

The cigar soliton has finite width equal to 2n. F irst, by taking F to be

the distance squared to t he origin, we see t hat w(gcig) s; 2n. Second, consider

any C^00 proper function F satisfying Lcig(F-^1 (c)) < oo for all c. Let D,N =

{z E !R^2 J N s; p(z) s; 2N}, which is approximately a round cylinder for N large.

Let CN be such that

Areacig( {F s; CN} n D,N) = Areacig( {F ~ CN} n D,N ).

Then Lcig(F-^1 (cN )) ~ Lcig(F-^1 (cN) nnN ). As N -too, a lower bound for the RHS

limits to 2n by an isoperimetric inequality for m anifolds with boundary.^7 Hence

W (F,gcig) ~ 2n.

Below, we shall use t he following Laplacians: D-euc = ::2 + ::2, D-cyl = ts', +

'2, and D. =Dog.

3.3. Ancient solutions with finite width.

In t his section we discuss the proof of the following result of Daskalopoulos and

Sesum.

THEOREM 28.41 (Classification of 2-dimensional noncompact ancient solutions

with finite widt h). If (M^2 ,g(t)) , t E (- oo,O), is a complete noncompact ancient

so lution to the Ricci flow with bounded positive curvature and finite width at each

time, then ( M^2 , g ( t)) must be a constant multiple of the cigar soliton. In particular,

the so lution extends to an eternal so lution.

The rest of the section is devoted to the proof of t his theorem. Recall the

following basic facts. Since Rg(t) > 0 and Mis noncompact, Mis diffeomorphic to

JR^2 and Area(g(t)) = oo for each t (in fact, t he areas of geodesic balls grow linearly

in terms of their radii). Furthermore, a consequence of the uniformization theorem

is that (M, g (t)) is conformal to t he plane for each t < 0 (see Theorem 15 in Huber

[151]).

Since Ricci fl.ow preserves the pointwise conformal class for 2-dimensional com-

plete solutions with bounded curvature, there exist global conformal coordinates

(x, y) : M -t JR^2 such t hat

g(x,y,t) = e-f(x,y,t)(dx^2 +dy^2 )

for some C^00 function f : M x ( -oo, 0) -t R Hence Rg = e f D-eucf. By the Ricci

fl.ow equation %tg = -Rg, we then obtain

(28.42)

of

at = R = D.gj.

Thus, the con formal factor u ~ e-f( x,y,t) satisfies gt ln u = Dog ln u; i.e., ftu =

Doeuc ln U.

The following inequality is a geometric basis for the proof of Theorem 28.41.

LEMMA 28.42 (Bochner-type inequality). Let (M^2 , g(t)), t E (-oo, 0), be a

complete noncompact ancient so lution with positive curvature. Suppose that f :

(^7) In the proof of the claim inside the proof of Lemma 28.43 below, we consider a simila r
situation in more d e tail.