- LI-YAU ESTIMATE FOR POSITIVE SOLUTIONS OF HEAT EQUATIONS 323
where L = log u, satisfies the heat-type equation
(25.67)
aP
87= b.P + 2 (\IL, \JP)+ 2 (1-c) l\l\JLl^2 - 2~j\li\ljL
+ 2 ((1 - c) ~j - ERij) \liL\ljL
+ (-2gii\JiRjk + \lkR - 2c\lkQ) · \lkL - D.Q.PROOF. Taking the time derivative of (25.64) while using (25.63) and
(25.65), we compute~: = (:Tb.) L + b. ( ~~) + 2c \ \l L, \l ( ~~) )
- 2ERij\liL\ljL
(25.68) = -2Rij\li\ljL + (-2gii\JiRjk + \lkR) · \lkL
D.P + D. ( (1 - c) I\/ Ll^2 - Q) + 2c (\/ L, \l P)
2c \IL, \l ((1-c) l\JLl^2 - Q) )-2ERij\liL\ljL.
Applying the standard Bochner-type formulaD.. l\l Ll^2 = 2 l\l\l Ll^2 + 2 (\/ L, \l (D.L )) + 2 Re(! L, \l L)
and the identity2 (1-c) (\IL, \l (D.L)) + 2c (1-c) \IL, \l l\JLl^2 ) = 2 (1-c) (\IL, \JP)
(which follows from \l P = \l D.L + c\l l\l Ll^2 ) to (25.68), we obtain (25.67)
by rearranging and combining terms. D
If we assume a lower bound on the Ricci curvature and bounds on Rij
and parts of its first two covariant derivatives, then we obtain the following.
COROLLARY 25.18 (Evolution inequality for the Harnack quantity). Sup-
pose that we have the bounds(25.69)
Re 2:: -K,
l-2gij\liRjk + \lkRI + i l\JQI:::; O',IRijl :::; A,
t:.Q :::; O'on M x [O, T] for some nonnegative constants K, A, and O'. If c E (0, 2/3),
then the Harnack quantity P satisfies the following heat-type inequality(25. 70)
where(25.71)
8P 2 - 3c 2 2- 8
2:: D.P + 2 (\IL, \JP)+ --(D.L) - 01 l\JLI - 02,
T nA2
and 02 ~ -+20'.
c