318 2S. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS('potential function' for the heat-type equation below) be a C^00 function
satisfying(25.50)on M x [O, T] for some constant C' < oo. Suppose that !sect (g (0))1 ::::; K
on M and suppose
(25.51) Re;::: -K and IR··l<A iJ -
on M x [O, T] for some finite and nonnegative constants K and A.
We have the following result.THEOREM 25.8 (Harnack estimate of Li-Yau-type for evolving metrics).
Let g (T), T E [O, T], be a family of Riemannian metrics on Mn as above
and let Q be as above. If
u: Bg(O) (p, 2 R) x [O, T] --+ lR+,
where R;::: l, is a positive solution to
OU
(25.52) OT = b.. 9 ( 7 )U - Qu,then for any EE (0, 2/3) we have the gradient estimate(25.53) ~ Olag u - (1-c) l\7logul^2 + Q;::: - n (^1 Cs C6 )
2 _ 3 E -;; + R + R2 + C1
in Bg(O) (p, R-Cn,K) x (0, T], where Cs depends only on n, K, A, T, and
SUPMx[O,T] l\7iRjkl, where C5 depends only on n, K, A, T, and E, where C7
depends only on n, K, A, E, and C', and where Cn,K E (1, oo) depends only
on n and K. In particular, taking R--+ oo, we have
(25.54) Ologu - (1-c) l\7logul^2 + Q;::: __ n_ (! + c1)
OT 2-3E T
in all of M x (0, T].In the case of the Ricci flow, we have the following.THEOREM 25.9 (Harnack estimate of Li-Yau-type for Ricci fl.ow). Let
g (T), TE [O, T], be a complete solution of the Ricci flow g 7 gij = -2~j on
a manifold Mn and let
Q : M x [O, T] --+ JR
be a C^00 function. Suppose there are constants K, C', and a radius R ;:::
max{l, )K} such that in Bg(T) (p, 2R) we have^2
(25.55) IRijl ::::; K, ~ IV7QI ::::; C', and b..Q::::; C'(^2) Note that in the cases where Rij = cRij, the inequality ~ [V'Qf ::::; C' is equivalent
to the first inequality in (25.50).