1547671870-The_Ricci_Flow__Chow

NOTES AND COMMENTARY 277

Hence by the Compactness Theorem (Section 3 of Chapter 7), there is a

subsequence (N^2 , hi ( t) , Xi) that limits to an ancient solution

(N^2 , h _ 00 (0) , X - 00 )

of the Ricci flow on a compact surface of positive curvature. By (9.13), the
scale-invariant entropy of the limit satisfies

E (h 00 (t)) =i--->oo Em E (hi (t)) =Em i--->oo E (h (ti+ R ( Xi, t ti ))) = E 00 ,

because limi__, 00 (ti+ t/ R (xi, ti)) = -oo for all t E (-oo, O]. In particular,
the entropy of the limit is independent of time. Thus by Proposition 5.40,
h 00 (t) is a shrinking round 2-sphere. Since constant-curvature metrics
minimize entropy among all metrics on 52 , we have E 00 ::; E (h (t)). But
since E (h (t)) is nonincreasing, E 00 2:: E (h (t)). It follows that E (h (t)) =
E_ 00 , hence that h (t) is a shrinking 2-sphere of constant curvature. 0

We now consider the case that the solution is Type II. Recall that we dis-
cussed backwards limits in Section 6 of Chapter 8. The self-similar solution
corresponding to the cigar soliton was introduced in Section 2 of Chapter 2.

PROPOSITION 9.24. Let (M^2 ,9(t)) be a complete Type II ancient solu-

tion of the Ricci flow defined on an interval (-oo, w), where w > 0. Assume

there exists a .function K (t) such that IRI ::; K (t). Then either 9 (t) is fiat

or else there exists a backwards limit that is the self-similar cigar solution.

PROOF. By Lemma 9.15, we have R 2:: 0. The strong maximum principle
implies that either R = 0 or else R > 0. From now on, we assume the latter.
By Corollary 9.21, the scalar curvature is pointwise nondecreasing. Hence
we have the uniform bound R::; K (0) on (-oo, O]. Now if M^2 is compact,
then Klingenberg's Theorem implies that its injectivity radius is bounded
from below by 7r / JK(O). If M^2 is noncom pact, we get the same bound
from Theorem B.65. So in either case, we can take a limit backwards in time
as in Section 6 of Chapter 8. Since the solution is Type II, we obtain an

eternal solution (M~, 900 (t)) satisfying 0 < R 00 ::; 1 and R (x 00 , 0) = l. By

Lemma 5.96, (M~, g 00 (t)) is isometric to the cigar solution (~^2 , 92:. (t)). 0

COROLLARY 9.25. Let (M^2 , g (t)) be a complete ancient solution defined

on (-oo, w), where w > 0. Assume that its curvature is bounded by some

.function of time alone. Then either the solution is fiat, or it is a round

shrinking sphere, or there exists a backwards limit that is the cigar.

Notes and commentary

The main references for this chapter are [58], [59], and [63]. The esti-

mate that the curvatures tend to positive in Section 3 was proved in Theorem

4.1 of [66]. Similar estimates, without the time term, were earlier proved

independently by Ivey [77] and Hamilton (Theorem 24.4 of [63]). Theorem