1547671870-The_Ricci_Flow__Chow

146 5. THE RICCI FLOW ON SURFACES

where the infimum is taken over all C^1 - paths ry : [t 1 , t2] ---+ M^2 joining x1
and x2. Then exponentiation of equation (5.39) yields a classical Harnack
inequality.

PROPOSITION 5.58. Let (M^2 , g (t)) be a complete solution of the nor-

malized Ricci flow with bounded positive scalar curvature. Then there ex-

ist constants C1 > 1 and C > 0 depending only on go such that for all

xi, x2 E M^2 and 0 :S t1 < t2,

• R --(x2 '-t2) > e-4 AC 1 erti -^1 > e-4-C(t2-t1) A
  R (x1, t1) - C1ert^2 - 1 -

(5.40)

10.2. The case that R (·, 0) changes sign. We now extend the dif-

ferential Harnack inequality to solutions of the normalized Ricci flow whose

curvature changes sign. Assume that r > 0. As in Section 8.2, we consider

the solution

of the ODE

r

s(t)=-----

1 - ( 1 - S: ) ert

d

- s = s(s - r)

dt

with initial condition

s(O) =so < Rmin(O) < 0.

We define

L = L (g, s) ~ log ( R - s).

A computation similar to that in Lemma 5.53 shows that L evolves by

(5.41) ot {) L A = b..L A + I \7 L A^12 + R - r + s.

Then we define

LEMMA 5.59. Let (M^2 , g (t)) be a solution of the normalized Ricci flow

on a compact surface with any initial metric such that r > 0. Then

{) A A ( A A) I A 1 1

2

otQ=b..Q+2 'VQ,'VL + 2 \7\7L+ 2 (R-r)g

(5.42) +sjvLJ

2

+(r- s)Q+s(R-r).