1547671870-The_Ricci_Flow__Chow

13. THE CASE THAT R (-, 0) CHANGES SIGN 153

COROLLARY 5.72. Let "ft be a smooth I -parameter family of loops in a

solution (M^2 ,g (t)) of the normalized Ricci flow. If 'YT is a weakly stable

geodesic loop, then

:t Lt( "It) I t=T ;:: ~ L T ('YT).

We are now prepared to show that the length of the shortest closed
geodesic is strictly increasing whenever it is small enough.

LEMMA 5.73. Let (M^2 ,g (t)) be a solution of the normalized Ricci flow

on a compact surface of positive Euler characteristic, and let Kmax (t) > 0
denote its maximum Gaussian curvature. Suppose 'YT is the shortest closed
g ( T )-geodesic at some time T > 0. If LT ('YT) < 27r / J K max ( T), then there
is E > 0 small enough so that for all t E ( T - E, T), there is a g ( t )-geodesic
"ft such that

Lt( 'Yt) < LT ('YT).

Note, however, that we do not claim the geodesics "It depend smoothly on

tE(T-E,T).

PROOF. By Corollary 5. 72, there is Eo > 0 sufficiently small so that

Lt ('YT) < LT ('YT) for t E ( T - Eo ,T). Choose points p and q on 'YT that divide

'YT into two segments 1J: and ry'f. of equal length with respect to the metric

g (t). These may not beg (t)-geodesic segments, but for some c E (O,i::o),

we have Lt ('Y~) < 7r / J K max ( t) for i = 1, 2 and all t E ( T - c, T). Thus by

Lemma 5.66, there exist unique g (t)-geodesics f31 and f32 between p and q

which are near ryJ: and ry'f. respectively. Then by the Gauss lemma, we have

Lt ({31) +Lt ({32) ::; Lt ('Y~) +Lt ('Y;) < LT ('YT).

Hence by Lemma 5.67, there is a smooth g (t)-geodesic loop 'Y with

Lt("() ::; Lt(f31) + Lt(f32) < LT ('YT).

We are now ready to prove the main result of this section.

D

PROOF OF PROPOSITION 5.65. Lemma B.34 in Appendix B (Klingen-

berg's Lemma) says that inj (M^2 , g ( t)) is no smaller than the minimum of

7r / J K max ( t) and half the length of the shortest closed g ( t )-geodesic. But by

Lemma 5. 73, the length of the shortest closed geodesic is strictly increasing

as long as it is less than 27r / J Kmax ( t). D

13. The case that R (-, 0) changes sign

In this section, we complete our proof of Theorem 5.1 for the case

of a compact Riemannian surface (M^2 ,go) such that x (M^2 ) > 0 and

Rmin (go) < 0. The road map for obtaining convergence in this case is

as follows. Armed with the entropy estimate of Proposition 5.44 and the