- STRATEGY FOR THE CASE THAT x (M^2 > 0) 131
As we saw in Section 4 of Chapter 2, the solution to (5.23) differs from the
solution to the normalized Ricci fl.ow
a
ot9 = (r - R) 9only by by the one-parameter family of diffeomorphisms lfJt generated by
the time-dependent vector fields \7 f (t). Since M^2 is compact, Lemma 3.15
implies that these diffeomorphisms exist as long as the potential function
f (t) does. And since the quantity IMl^2 is invariant under diffeomorphism,
the estimate IMl^2 :S Ce-ct will hold on the solution to the modified Ricci
fl.ow (5.23). Moreover, as we did for the curvature in Sections 5 and 6, one
can obtain estimates for all derivatives of M, which prove that the solution
9 (t) to the modified fl.ow converges exponentially fast in all Ck to a metric
900 such that M 00 vanishes identically. By equation (5. 7), this will imply that
900 is a gradient soliton. Then Proposition 5. 21 (which uses the Kazdan-
Warner identity, hence the Uniformization Theorem) will imply that 900 is
a metric of constant positive curvature. It will then follow that there exist
positive constants ck, Ck for each k E N such that the solution 9 (t) of the
modified fl.ow satisfies
lvk RI ::; Cke-ckt.
But by diffeomorphism invariance, the same estimates must hold for the
solution of the unmodified fl.ow. In this way, we will be able to conclude
that the normalized Ricci fl.ow starting at a metric of strictly positive scalar
curvature converges exponentially fast to a constant curvature metric.
In order to obtain uniform positive bounds for R, we proceed as follows.
We begin by developing two important technical tools.
STEP 1. The surface entropy of a compact 2-manifold ( M^2 , 9 ( t)) of
positive curvature evolving by the normalized Ricci fl.ow is defined asN (9 ( t)) ~ { R log R dA.
}M2
In Section 8, we will prove that N (9 ( t)) is strictly decreasing unless 9 ( t)
is Einstein; we shall later use this result to obtain uniform bounds on the
scalar curvature of a solution to the normalized Ricci fl.ow on a surface of
positive Euler characteristic. (If R (0) changes sign, there is a modified en-
tropy formula; in this case, in this case, we will prove that N (9 (t)) remains
bounded.)
STEP 2. In Section 9, we obtain uniform bounds for the metric, the
scalar curvature, the gradient of the scalar curvature, and the diameter of a
solution on a surface of positive Euler characteristic. In particular, we show
that if I RI :S "" on a time interval [to, to + 1 / ( 4"")], then I \7 RI :S 4""^3 /^2 on the
same interval. Together with the entropy result obtained in Step 1, we use
this to argue that IRI is uniformly bounded independently of time, and that
the diameter of the solution is bounded. In this step, the positivity of the