1547671870-The_Ricci_Flow__Chow

138 5. THE RICCI FLOW ON SURFACES

9.1. Bounds for the metric on a surface with x (M^2 ) > 0. Define

Rmin (t) ~ minxEM2 R (x, t) and Rmax (t) ~ maxxEM2 R (x, t).

LEMMA 5.45 (Doubling-time estimate). Let (M^2 ,g (t)) be any solution

of the normalized Ricci flow on a closed surface with r > 0. Then for any
to E [O, oo), the estimate

Rmax (t) :S: 2Rmax (to)

holds for all x E M^2 and

t E [to, to+ 2 Rm: (to)].

PROOF. By (5.3), the scalar curvature evolves by

8 2

ot R = .6.R + R - r R.

Since r > 0, we have Rmax (t) > 0 for all time. So at a maximum in space,

ut ~ R < - .6.R + R^2.

The solution of the initial value problem

dp 2

dt = P , P (to) = Rmax (to)

is

1

P (t) = R;;;lx (to)+ to - t"

Hence the maximum principle implies that for to :::::; t:::::; to+ 1/2Rmax (to),

Rmax (t) :S: 2Rmax (to)·

0

REMARK 5.46. This is a prototype of the general Doubling-time es-

timate derived in Corollary 7.5.

The lemma implies that the metrics are uniformly equivalent in the same

time interval. In particular, a lower bound for the metric is readily obtained.

COROLLARY 5.47. Let (M^2 ,g(t)) be any solution of the normalized

Ricci flow on a closed surface with r > 0. Given any to E [O, oo), the

estimate

1

g (x, t) 2: - g (x, to)

e

holds for all x E M^2 and t E [to, to+ 2 Rm~x(to)] ·

PROOF. At any x E M^2 , we may write the metric as

(5.31) g ( x, t) _ - e ftt o (r -R(x,r)) dr g ( x, t o. )