1547671870-The_Ricci_Flow__Chow

6. TYPE I ANCIENT SOLUTIONS ON SURFACES 275

whenever the solution exists for t E [-a, w). When the solution is ancient,
letting a --+ oo gives the result. D

In the successor to this volume, we will also prove the following state-

ment.

LEMMA 9.22. If (N2, h ( t)) is a complete ancient Type I solution of the

Ricci flow with strictly positive scalar curvature, then N2 is compact.

Assuming this for now, we will conclude this volume by characterizing

all complete ancient solutions of the Ricci flow on surfaces. We begin with

a classification of all complete ancient Type I 2-dimensional solutions.

PROPOSITION 9.23. A complete ancient Type I solution (N2, h (t)) of

the Ricci flow on a surface is a quotient of either a shrinking round 52 or

else a fiat IR^2.

PROOF. Since (N2, h ( t)) is an ancient Type I solution, it exists on a

maximal time interval ( - oo, w) containing t = 0 and satisfies a curvature

bound of the form

c

IR (x, t) I :S Cn IRm (x, t) I :S Ttf

for all x E M^3 and t E (-oo, 0). By Lemma 9.15, R 2: 0 on N2 x (-oo, w).

By the strong maximum principle, (N2, h ( t)) is either flat and hence a

quotient of IR^2 or else has strictly positive scalar curvature. From now on,

we assume the latter.

By Lemma 9.22 (below) N2 is compact, hence diffeomorphic to either

52 or IRIP^2. By passing to the twofold cover if necessary, we may assume

N2 ~ 52. The area A of N2 evolves by

~A= - { RdA = -47rX (N2),

dt }N2

where x (N2) is the Euler characteristic. Since A (t) --+ 0 as t--+ w by [60],

we have

A (t) = 47rX (N2) · (w - t) = 87r (w - t).

Recall (Section 8 of Chapter 5) that the entropy

E (h (t)) ~ { Rlog (RA) dA

}N2

= { Rlog [R (w - t)] dA + 47rX (N^2 ) log [47rX (N^2 ) ]

JN^2

is a scale-invariant and nonincreasing function of time. Because (N2, h ( t))

is an ancient Type I solution, we have

sup R (x, t) · (w - t) ::; C < oo,

N2x(-oo,w)