1547671870-The_Ricci_Flow__Chow

168 5. THE RICCI FLOW ON SURFACES

IR (xi, ti) I maxM2 IR(., ti) I and consider the sequence of parabolically
dilated solutions

(5.55)

defined for -tjR(xj,tj) ::::; t < (T-tj)R(xj,tj)- The dilated solutions

(M^2 , gj (t)) satisfy a curvature bound which depends on t but is uniform in

j. Indeed, if we are dilating about a Type I singularity, there exists

such that

n ~ limsup (T-t) Rmax (t) < oo

t/T

n

R[gj(t)]::::; n-t;

whereas if the original singularity is Type Ila, there is a sequence nj / oo
such that
1
R [gj (t)] ::::; 1 - t/Oj

Since by Corollary 5.95, there is a uniform positive lower bound for all

inj (M^2 , gj (0)), we can apply the Compactness Theorem from Section 3 of

Chapter 7 to obtain a pointed subsequence (M^2 ,gj (t) ,xj) that converges

uniformly on any compact time interval and in every Ck-norm to a complete

pointed solution

(M~, 900 (t), Xoo)

of the Ricci flow with scalar curvature R 00 • In the case of a Type I limit,

g 00 (t) is an ancient solution that exists for -oo < t < n and satisfies the

curvature bound

n

Roo::::; Q-(

In the case of a Type Ila limit, 9= (t) is an eternal solution that exists for

-oo < t < oo and satisfies

Roo (·, ·) ::::; 1 = R 00 (x 00 , 0).

The key step toward our goal will be to show that a Type Ila limit

cannot occur. This is a consequence of the fact that an eternal solution

with strictly positive curvature that attains the space-time maximum of its

scalar curvature must be a steady Ricci soliton. In the particular case of a

surface, the result is the following result.

LEMMA 5.96. The only eternal solution (N^2 , g (t)) of the Ricci flow on

a surface of strictly positive curvature that attains its maximum curvature

in space and time is the cigar (JH.^2 , 92: ( t)).

PROOF. We claim that (N^2 , g (t)) is a gradient Ricci soliton. The result

follows from this claim and Lemma 2.7, which proves that the only gradient

soliton of positive curvature in dimension n = 2 is the cigar.