1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

2. LI-YAU ESTIMATE FOR POSITIVE SOLUTIONS OF HEAT EQUATIONS 31g

for each TE [O, T]. If

u: U Bg( 7 ) (p, 2R) x {r}-+ IR+

TE[O,Tj

is a positive solution to

then for each TE [O, T] we have in Bg( 7 )(P, R)

(25.56) --alog u - (1-c-) IV'logul^2 +Q 2: ---n ( -^1 + - + - + Cs Cg C10 )

or 2 - 3c- T R R^2 '

where Cs depends only on n and K, where Cg depends only on n and E, and

where C10 depends only on n, K, E, and C'.

REMARK 25.10. A difference in the above two theorems is that, in the
latter case, the bounds in the hypothesis are assumed only on a ball instead
of on the whole manifold.

PROBLEM 25.11. Prove a Li-Yau-type differential Harnack estimate for

positive solutions of heat-type equations coupled to the backward Ricci flow

using only local bounds on the curvature and potential function.

2.2. Some consequences - classical-type Harnack estimates.
A standard integration along paths of the gradient estimate (25.54)
yields the following 'classical-type' Harnack inequality, which compares the
solution at different points in space-time.

COROLLARY 25.12 (Classical-type Harnack inequality for evolving met-

rics). Assume that g is a complete metric on Mn such that for each T E

[O, T],

(25.57)

on M, where Co < oo. Then under the assumptions of Theorem 25.8 we

have that for any x1, x2 E M and 0 < 71 < 72 :S T,