328 25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICSTo obtain upper bounds for 1v:12
and ~~ - bi.¢, evidently it sufficesto obtain upper bounds for [\7 f[^2 and -gt+ b.f. Combining (25.87) and
(25.88), we haveo</> - bi.¢+ (2 + n ) IY' 1>12
OT 2 (2 - 3c) E </>(25.89) ::::; ~max VC¢ · { 0, - OT+ of b.f } + ( 3 + n ) c^2
2 ( 2 _ 3c) E R2 IY'fl ·
Note that the differential Harnack estimate of Theorem 25.8 shall follow
from (25.82), which in turn relies on showing (25.79). In view of (25.89),
the desired estimate (25. 79) shall follow from upper bounds for [\7f1^2 and- gt + b.f; we prove this in the next subsection.
T 2
There are a few ways to try to approach the upper bounds for [\7 fl and
-gt+ b.f:
(1) Define f ( ·, T) to be the distance function top at time 0 (then</> is
independent of time).(2) Define f ( ·, T) to be the distance function top at time T.
(3) Define f ( ·, T) to be a smoothing of the distance function at time
0, so that one has Hessian bounds, which in turn controls the time-
dependent Laplacian Ll of f.
An issue with (1) is that the consideration of controlling b. = b.g(r)fby b. 9 (o)f and gr (b.f) yields a term which is -Rij'Vi'Vjf (see (25.65));
however it is difficult to obtain two-sided bounds for the Hessian of the
distance function. We do not consider this method.An issue with (3) is that one construction of such a function f (see §4 of
Chapter 26 in this volume) appears to use global bounds (on all of M) for
the curvature whereas we wish to use only curvature bounds in a ball. In
the next subsection we consider this method for the case of a general family
of metrics.
An issue with (2) is to estimate the heat operator of the time-dependent
distance function. In view of Perelman's changing distances estimate (The-
orem 18. 7), in the subsection after the next, we shall consider this method
for the case of a solution to the Ricci fl.ow.2. 7. Completing the proof of the Li-Yau inequality in the case
of a general family of metrics.
Recall that in the hypothesis of Theorem 25.8 we have assumed the
bound [sect (g (0))1 ::::; Kon M. By Proposition 26.49 below, we may definef : M --t JR. in (25.85) to be a C^00 function such that
(1) (distance-like)
(25.90) dg(O) (x,p) + 1 :Sf (x) :S dg(O) (x,p) + Cn,K