1547671870-The_Ricci_Flow__Chow

148 5. THE RICCI FLOW ON SURFACES

and hence reach the conclusion. D

This is our differential Harnack inequality in the case that the curvature
changes sign. As in the case of strictly positive curvature, we can integrate
to obtain a classical Harnack inequality.

PROPOSITION 5.61. Let (M^2 , g (t)) be a solution of the normalized Ricci
flow on a compact surface. Let go be any initial metric such that r > 0.
Then there exists a constant C > 0 depending only on go such that for any
x1,x2 EM^2 and 0 :S t1 < t2,

(5.43)

11. Convergence when R (-, 0) > 0

We are now ready to prove Theorem 5.1 in the case the R (-, 0) > 0.

The first step is to apply the LYH differential Harnack inequality to obtain

a uniform positive lower bound for R.

PROPOSITION 5.62. Let (M^2 , g (t)) be a solution of the normalized Ricci

flow on a compact surface such that R [go] > 0. Then there exists a constant
c > 0 depending only on go such that
R(x, t) 2: c > 0

for all x E M^2 and t E [O, oo).

PROOF. Observe that the differential Harnack inequality implies that

for any points xi, x2 E M^2 and times 0 :S ti < t2, one has

R (x2, t2) > e-1-C(t2-t1)

R(xi,ti) - '

where

1

A= A (xi, t1, x2, t2) ~ i~f ti t2 Id d~^12 dt.

Notice that R 2: ce-r for times 0 :S t :S 1, and consider any point and

time (x, t) with t 2: 1. Choose any xi E M^2 such that r :S R (xi, t - 1) :S

Rmax (t - 1). Then we have

R (x, t) 2: e-1-c R (x 1 , t - 1) 2: re-c. e-A/^4.

Thus to obtain a uniform lower bound for R, it will suffice to obtain a

uniform upper bound for A (xi, t - 1, x, t).

It follows from Proposition 5.51 and formula (5.31) that

e-C g (t - 1) :S g (T) :Ser g (t - 1).