1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

3. BASIC SINGULARITY THEORY FOR RICCI FLOW 469

solution or sausage model (see [311] or [141]) of the Ricci fl.ow is the

metric g = u · h defined for t < 0 by

>.-^1 sinh (->.t)

(Vl-^2.^22 ) u (x, t) = cosh x + cosh >.t'

where>.> 0.

LEMMA A.46 (Rosenau solution and its backward limit). The metric

defined by (Vl-2.22) fort < 0 extends to an ancient solution with positive

curvature of the Ricci flow on 52. The Rosenau solution is a Type II ancient

solution which gives rise to an eternal solution if we take a limit looking

infinitely far back in time. In particular, if one takes a limit of the Rosenau

solution at either pole x = ±oo as t ---+ -oo, one gets a copy of the cigar

soliton.

Note that the sausage model is an ancient Type II solution which en-
counters a Type I singularity.
3.5.2. Classification results. The following provides a characterization
of the cigar soliton. (See Lemma 5.96 on p. 168 of Volume One.)

LEMMA A.47 (Eternal solutions are steady solitons, 2d case). The only

ancient solution of the Ricci flow on a surface of strictly positive curva-

ture that attains its maximum curvature in space and time is the cigar

(IR^2 ,g~ (t)).

The following classifies 2-dimensional complete ancient Type I solutions.

(See Proposition 9.23 on p. 275 of Volume One.)

PROPOSITION A.48 (Nonfl.at Type I ancient surface solution is round

52 ). A complete ancient Type I solution (N^2 , h ( t)) of the Ricci flow on a

surface is a quotient of either a shrinking round 52 or a fiat IR^2.

We have the following result for 2-dimensional Type II solutions. (See
Proposition 9.24 on p. 277 of Volume One.)

PROPOSITION A.49 (Type II ancient solution backward limit is a steady,

2d case). Let (M^2 , g (t)) be a complete Type II ancient solution of the Ricci

flow defined on an interval (-oo, w), where w > 0. Assume there exists a

function K (t) such that IRI :S K (t). Then either g (t) is fiat or else there
exists a backwards limit that is the cigar soliton.

Combining the above results, we obtain the following. (See Corol-
lary 9.25 on p. 277 of Volume One.)

COROLLARY A.50 (Ancient surface solutions). Let (M^2 ,g (t)) be a com-

plete ancient solution defined on (-oo, w), where w > 0. Assume that its

curvature is bounded by some function of time alone. Then either the solu-

tion is fiat or it is a round shrinking sphere or there exists a backwards limit

that is the cigar.