- LI-YAU ESTIMATE FOR POSITIVE SOLUTIONS OF HEAT EQUATIONS 321
on Bg(T) (p, R), where g is some complete metric on M and Co < oo. Sup-
pose that x1, x2 E M are such that there exists a minimal geodesic 'Y joiningx1 and x2 with respect to (M, §) with/ ( r) E Bg(T) (p, R) for each r E [O, T].
~ '.
P = u Bg(T) (p, 2R) x {r}.
TE[O,Tj
Then for a positive solutionu: P-+ IR+
to (25.52) and 0 < r1 < r2 ::; T we have
u(x2,r2) > e-G12(T2-T1) (r2)-2..'.'3e exp (- Co d~(x1,x2))
u(x1, r1) - r1 4(1 - c) r2 - r1 'where C12 = ~ + ~§ + C10 +supp Q.
EXERCISE 25.14. Prove the above corollary.
As a special case of Corollary 25.12,
(25.59).u(x1, r1) ::; e C^11 (^72 -^71 )+__9i__^4 <^1 -el (r2) 2..'.'3e
71u(x2, r2) for x2 E Bg (x1, Jr2 - r1).Integrating this with respect to x2 over the ball, we have
(25.60)u(x1, r1) ::;^012 (^72 )2
..'.'3
e { u(x, r2)dμg (x),Volg Bg (x1, Jr2 - r1) r1 J B 9 (x 1 ,.JT 2 -T 1 )
where C12 = eGu(Tz-Ti)+^4 <f_gc:l. Since x2 E Bg (x1, Jr2 - r1) is equivalent to
x1 E Bg (x2, Jr2 - r1), integrating (25.59) with respect to x1 over a ball, we
obtain
(25.61)
u(x2,r2) ~ c;:} (Tl)
2
_'.'3
e r u(x,r1)dμg (x).VolgBg(x2,Jr2-r1) r2 }B 9 (x 2 ,.JT 2 -T 1 )
The above inequalities (25.60) and (25.61) may be considered as mean
value-type inequalities. Compare them with the mean value inequality bf §1
of this chapter, proved by Moser iteration, where the integral on the RHS ()f
the inequality is over a parabolic cylinder instead of a spatial ball at a given
time.
REMARK 25.15. In view of Perelman's differential Harnack estimate for
the adjoint heat kernel, we are interested in obtaining bounds for this adjoint
heat kernel under local hypotheses. We shall obtain upper and lower bounds
for the heat kernel using the Li-Yau inequality and other estimates and then
use symmetry (see Lemma 26.3) to obtain upper and lower bounds for the
adjoint heat kernel. One also has gradient estimates as discussed in §2 of
Chapter 16 and §4 of Appendix E both in Part IL