1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

320 25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS

PROOF. Let 'Y: [Ti, T 2 ] --+ M be a smooth path joining x1 to x2. By the

fundamental theorem of calculus,

1

u(x2, T2)

og u(x1, T1)

1

T2 d

= -d (log U ( / ( T) , T)) dT

Tl T

= 1T

2

( ( 8l~g u) ( 'Y ( 7 ) , T) + / (V' log u) ( 'Y ( T) , T) , ~'Y ( T)) ) dT.

Tl \ UT \ T g(T)

Applying (25.54), we have

1

u(x2, T2)

og u(x1, T1)

;:::: 1T

2

((1 -E) [V' log u[~(T) ( 'Y ( T) , T) + / (V' log u) ( 'Y ( T) , T) , ~'Y) ) dT

Tl \ T g(T)

• 1:

2

( ( 2 _ ~E) T - Q ( / ( T) , T) - C7) dT

2:- 1 1T2ld'(T)l2 dT--n-log(T2)-c11(T2-T1),

4(1-E) Tl dT g(T) 2-3E Tl

where Cn = C7 +supMx[T 1 ,T 2 ] Q. Exponentiating, we obtain for any path/

u(x2,T2) > e-C11(Tz-T1) (T2)-2.:'..'3e e-4f1<=~)

u(x1, T1) - T1 '

where

A (r) = 1T2 I ~'Y (7)12 dT.
Tl T g(T)
Now let 'Y be a constant speed minimal geodesic with respect to g, so that

I

d/ (T)I = d9 (x1,x2).

dT g T2 - T1

Then, using (25.57), we obtain

A (r) ~Co 1T2 I d1 (T)l2 dT =Cod~ (x~, x2).

Tl dT g T2 Tl

The corollary follows. D

In the case of the Ricci fl.ow, by the same argument as in the proof of
Corollary 25.12, we have the following.

COROLLARY 25.13 (Classical-type Harnack inequality for Ricci fl.ow).
Let g (T), TE [O, TL be a complete solution of the Ricci flow as in Theorem
25.9 and such that for each TE [O, TL

C 01 g ~ g (T) ~Cog