1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

306 25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS

where the volume is measured with respect to the metric g + dt^2 •

The time-dependent setup is as follows. Let g be a complete C^00 metric

on a smooth manifold Mn and suppose that n is a compact domain in M

with C^00 boundary &n. (The boundary &n may be empty. For example, if

Mis closed, we may taken= M.) Let g (T), TE [O, T], where TE (0, oo),

be a C^00 family of C^00 metrics on n such that the initial metric satisfies the

bounds

(25.1)

inn for some constant Co E [1, oo). Sometimes we shall take g = g (0), in

which case Co = 1.

Define the time-dependent symmetric 2-tensor Rij by

&

(25.2) OT9ij ~ 2Rij·

Let

(25.3) n (x, T) ~ gij (x, T) Rij (x, T).

A special case we shall later consider is where Rij = Rij is the Ricci tensor

(so that g (T) is a solution to the backward Ricci flow). Let

(25.4) A~ sup IRij (x, T)/ 9 ( 7 ) < oo.

nx[O,T]

By (25.1) and (25.2), we have

(25.5) cr;^1 9:::; g (T) :::; Cog

in n x [O, T], where Co ~ Coe^2 AT < 00.

Let

Q : n x [O, T] -+ IR

be a C^00 function and let u: n x [O, T] -+IR+ be a positive subsolution to

&u

(25.6) OT :::; b..g(T)U - Q u.

When Rij = ~j, we are most interested in solutions to the adjoint heat
equation g~ = b.. 9 ( 7 )u - Ru.
Define the parabolic cylinder
(25.7) P9 (x,T, r, -r^2 ) ~ B9 (x, r) x [T - r^2 , T]
based at the point (x, T) with radius r.
The following is a version of Moser's parabolic mean value inequality for
heat-type equations with respect to evolving metrics.

THEOREM 25.2 (Parabolic mean value inequality). In the above setup,

suppose that we have the Ricci curvature lower bound Re (g) 2': -K in n,

where K 2': 0, and let

u : :11 x [O, T] -+ IR+