- AN ALTERNATIVE STRATEGY FOR THE CASE x (M^2 > 0) 165
PROOF OF THEOREM 5.88. We shall show that for any E > 0, we have:t CH (M2' g (t)) lt=to 2: -E
at all times to such that CH (M^2 , g (to)) < 47r. Given such a time t 0 , there
is by Lemma 5.85 a smooth embedded loop 'Yo whose isoperimetric ratio
measured with respect tog (to) satisfies the identity
Since 'Yo is a minimizer of CH at time to, we have the relations
(:p log CH) (0, to) = 0
( :; 2 log CH) (0 , to) 2: 0.
Using these and the inequality 47r - CH 2: 0, we conclude from Lemma 5.92
that
( :t log CH) (0, to) 2: 0.
This is possible only if for each E > 0 , there exists o > 0 depending on E
such that for all t E (to - o, to), one has
log CH (0, t) ::::; log CH (0, to)+ E (to - t) =log CH (M^2 , g (to)) + E (to - t).
Hence for all t E (to - o, to), we have
log CH (M^2 , g (t)) +Et::::; log CH (M^2 , g (to)) + Eto.
D
7. Strategy for the case that x (M^2 > 0)
We shall now complete the alternative strategy for proving convergence
of the Ricci fl.ow on a surface of positive Euler characteristic such that R ( ·, 0)
might change sign. This approach uses the the isoperimetric estimates de-
rived in the previous section. It also uses the entropy estimate of Section 8
and the L YH differential Harnack estimate of Section 10 for metrics of pos-
itive curvature, but (perhaps surprisingly) not their extensions to the case
of variable curvature. It also relies on other important methods that will be
developed later in this volume. In particular, the approach uses techniques
of dilating about a singularity. (We will study such techniques in Chapter
8.)REMARK 5.93. The reader will find related results in Section 6 of Chap-
ter 9.