1547671870-The_Ricci_Flow__Chow

156 5. THE RICCI FLOW ON SURFACES

As in Proposition .5.62, the desired positive lower bound for R - s will follow
from (5.46) once we bound A from above. Recall the integral formula (5.31)
for the metric, which implies that for any T 2: t - 1, one can write

g (x, T) = g (x, t - 1) exp r (r - R) df.

lt-1

Using the consequence of Lemma 5.9 that R 2: Rmin (0) and the uniform
upper bound for R obtained in Lemma 5. 74, it follows from this formula that

there exist c 1 < C 1 independent of x E M^2 such that for all T E [t - 1, t],

one has

c 1 · g (x, t - 1) ~ g (x, T) ~ C2 · g (x, t - 1).

As in the proof of Proposition .5.62, this estimate implies that there exists
C2 > 0 such that

A~ C2 [dist g(t) (x, x1)]2,

whence the desired upper bound for A follows from the diameter bound of
Lemma 5.75.
Now by substituting the uniform upper bound for A into estimate (5.46)
and using the fact that x E M^2 is arbitrary, one concludes that there exists

c > 0 such that

Rmin ( t) = inf R ( x, t) 2: E: + s ( t).

xEM^2

Since s (t) / 0 as t----+ oo, this proves the lemma. D

Once the curvature becomes strictly positive, we may proceed exactly
as in Section 11. This completes our proof of the following result.

THEOREM 5.77. Let (M^2 ,go) be a closed Riemannian surface with aver-

age scalar curvature r > 0. The unique solution g (t) of the normalized Ricci

fiow with g (0) = go converges exponentially in any Ck-norm to a smooth
constant-curvature metric g 00 as t----+ oo.

14 Monotonicity of the isoperimetric constant

In the remainder of this chapter, we illustrate an alternative strategy
for proving convergence of the Ricci fl.ow starting from an arbitrary initial
metric on a surface of positive Euler characteristic. The basic idea is as
follows. A solution of the unnormalized Ricci fl.ow with r > 0 will become
singular in finite time. If one can estimate the injectivity radius of such
a solution on a scale appropriate to the curvature and if one can rule out
slowly-forming singularities, one can then show that a sequence of parabolic
dilations formed at the developing singularity will converge to a round sphere
metric. This is called a blow-up strategy, because one 'blows up' the solution
at the developing singularity. Blow-up strategies are extremely important in
understanding singularity formation of the Ricci fl.ow in higher dimensions.