1547671870-The_Ricci_Flow__Chow

NOTES AND COMMENTARY 171

by (5.52) and (5.53), we have

R(-,[) =r(T-t) R(-,t).

But since g (t) has a Type I singularity, the right-hand side is uniformly
bounded:
limsup (T - t) Rmax (t) ~ D < oo.
t/T

Now we finish the argument.

D

PROPOSITION 5.99. If (M^2 , go) is a closed Riemannian surface with

average scalar curvature r > 0, then the unique solution g (t) of the unnor-

malized Ricci flow with g (0) = go encounters a singularity at some finite

time T. A sequence (M^2 ,gj (t)) of parabolic dilations formed as tj / T

converges locally smoothly to a shrinking round sphere ( 52 , g= ( t)).

PROOF. We know that (M^2 , gj (t), Xj) converges locally smoothly in

the pointed category to a limit (M~,g= (t) ,x=) that must be an ancient

solution with bounded curvature. By Lemma 9.15, its scalar curvature is
positive. Let T ~ -t. Then by Proposition 5.39, the entropy N (g (T)) is
increasing in T and is bounded above by the entropy of a soliton. Thus
(going backwards in time) we can take a limit as T --> oo to get a solu-
tion (M§=, g2= (t), x2 00 ) which has constant entropy, hence must be a soli-
ton. By Proposition 5.21, (M§ 00 ,g2 00 (t) ,x2 00 ) is a shrinking round sphere.
But its entropy is minimal. Since N (g ( T)) cannot decrease as T increases,
this implies that N (g ( T)) must be constant, hence that the original limit
(M~, g 00 (t), x 00 ) was itself the shrinking round sphere. D

REMARK 5.100. The assertion that the limit (M~,g 00 (t) ,x 00 ) must be
the shrinking round 2-sphere also follows from Proposition 9.23.

Notes and commentary

The main reference for this chapter is Hamilton's paper on the Ricci

flow on surfaces [60]. Unless otherwise noted, the results in this chapter

may found there. In particular, the methods of entropy and Harnack esti-

mates, Ricci solitons, and potential of the curvature were first introduced to

the Ricci flow in [60]. Curvature derivative estimates were established by

Shi in [117, 118]. Originally, curvature derivative estimates were obtained

in Corollary 8.2 of [60] by bounding the L^2 -norms of all derivatives and ap-

plying the Sobolev inequality. (See also Theorem 13.4 in [58].) The method

of proof we give for the upper bound of R in Proposition 5.30 follows [30].

The direct proof of the entropy estimate of Proposition 5.39 was given in

[29]; the results in subsection 8.1 are also from there. The method of prov-

ing the differential Harnack inequality in section 10 was first used by Li and

Yau [92] for solutions of the heat equation. (See Section 6 of [60].) The

extension to the case where R changes sign in Section 10.2 was _established