1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

2. LI-YAU ESTIMATE FOR POSITIVE SOLUTIONS OF HEAT EQUATIONS 325

point where ¢ =I= 0

~!!__ (T</>P) = ¢ oP + 0¢ p +</JP

T OT OT OT T

2: ¢ ( b.P + 2 (\7 L, \7 P) +

2

~

3

s (b.L)^2 - C1 l\7 Ll^2 - C2)

+ 0¢ P+ </JP

OT T

(25.73) = b. (¢P) + ( ~~ - b.¢) P - 2 ( ~¢, \7 (¢P))

+ 21\7:1

2

P + 2 (\JL, \7 (¢P)) - 2P (\7 L, \7¢)

+

2

-

3

s ¢ (b.L)^2 - C1¢ l\JLl^2 - C2¢ + P,
n T
where C1, C2 2: 0 are as in (25.71).
We now apply the weak maximum principle to our calculation. By
(25.72), the support of T</J (x) P (x, T) is contained in B(p, 2 R) x [O, T]. Let
1' E (0, T] and suppose that T</JP is negative somewhere in B(p, 2R) x [O, 7].
(Otherwise we have the estimate T</JP 2: 0 in M x [O, 7'].) Then there exists
a point

(xo, To) E B(p, 2 R) x (0, 7']

at which T</JP attains a negative minimum. We shall obtain a lower bound
for T</>P at this negative minimum.
At (xo, To) we have ¢ =I= 0 and

(25.74) \7(¢P) = 0, b.(¢P) 2: 0,

0

and OT ( T</>P) ::::; 0.

The rest of the calculations in this subsection occur at (xo, To).
Substituting these inequalities into (25.73), we have

(25.75) O 2: ( ~~ - b.¢) P + 2 J\7:1

2

P - 2P (\7 L, \7¢) +

2

~

3

s ¢ (b.L)^2

- C1¢ J\JLJ^2 - C2¢ +</JP.

TO

Since by (25.64)

(25.76) (b.L)^2 = (P-sJ\JLJ^2 )2 =P^2 -2sPl\JL[^2 +s^2 [\JLJ^4

and since for any 5 > 0

(25.77)