1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

4. APPLICATIONS OF HAMILTON'S COMPACTNESS THEOREM 145

On the other hand, CKk^8 R (gk)-^8 ---+ 0 · R (g 00 )-^8 = 0. Hence we conclude

that Re (g 00 ) = iR (g 00 ) g 00 on the subset of M 00 where R (g 00 ) > 0. On the

other hand, the contracted second Bianchi identity implies R (g 00 ) = const

in any connected subset of the set where R (g 00 ) > 0. Hence we conclude

R (g 00 ) = 1 on all of M 00 , so that Re (g 00 ) = kg 00 on M 00 • D

4.3. Ricci flow on closed surfaces with x > O. Now we give various

proofs, which are variations on a theme, of the following consequence of

Theorem 5.77 on p. 156 of Volume One.

THEOREM 3.31. If (M^2 ,go) is a closed Riemannian surface with posi-

tive Euler characteristic, then a smooth solution g (t) of the Ricci flow with

g (0) = go exists on a maximal time interval [O, T) with T < oo. More-

over, there exists a sequence {(xk, tk)} with tk ---+ T such that gk (t) ~

Rkg (tk + Rk^1 t) , with Rk = R (xk, t~), converges to a solution (M^2 , g 00 (t))

with constant positive curvature.

First we recall the ideas of some proofs from Volume One. The first

proof relies on the monotonicity of the quantity I \i' i \i' j f -! /::i..f gij I 2.

PROOF #I. In this proof, which is the original proof of Hamilton, we

actually recall the exponential convergence (not just sequential convergence)
in C^00 of the normalized flow. Consider the normalized flow gtg = (r - R) g
on the maximal time interval [O, T). In the rest of this proof we abuse no-
tation by using g (t), t E [O, T), to stand for the normalized solution rather
than the unnormalized solution in the statement of the theorem. One can
prove that T = oo. If R (g (to)) > 0 for some to< oo, then we may combine
the entropy and Harnack (or Bernstein-Bando-Shi derivative) estimates to

show that there exist c > 0 and C < oo such that

O<csRsC

on M x [to, oo). Let Mij ~ \i'i\i'jf - !t:i..fgij, where /::i..f .;. R-r. From the
equation (see Corollary 5.35 on p. 130 of Volume One)

~ IMl2 = t:i.. IMl2 - 2 l\i' Ml2 -2R IMl2

8t

S !::i.. IMl^2 - 2c IMl^2

(using the lower bound for R), we have

(3.13) IMI S C1e-ct

for some C 1 < oo. We also have for p EN (see Corollary 5.63 on p. 149 of
Volume One)

(3.14) l\i'P Ml S Cpe-cpt

for some cp > 0 and Gp < oo. By the diffeomorphism invariance of the

estimates (3.13) and (3.14), this implies that the modified equation

8

otg = (r - R) g + L\Jfg