376 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS
Hence, if u ~ 0, then
I(r2) -I(r1) :S -1~
2
r~l lr 'I/Jr (:t -.6) udμg(t)dtdr.As an immediate consequence, we have the following.COROLLARY 26.45. Let (Mn,g(t)) be a complete solution to the Ricciflow. If u is a nonnegative supersolution to the heat equation, then
(26.122)for 0 < ri < r2. Moreover, for r > 0
(26.123)Note that inequality (26.123) follows from taking ri ---+ 0 in (26.122)
since limr 1 -to I(r1) = u(xo, to).
COROLLARY 26.46. Let (Mn, g(t)) be a complete solution to the Ricciflow. If w(x, t) is a supersolution to the heat equation which attains its
minimum over M x [O, to] at a point (xo, to) and if the scalar curvature is
nonnegative, then w(x, t) = w(xo, to) for all x EM and t::::; to.
PROOF. Applying (26.123) to u(x, t) ~ w(x, t)-w(xo, to), we obtain for
all r > 0
{ IV''l/Jrl^2 w dμg(t)dt::::; { (IV''l/Jrl^2 + 'l/JrR) w dμg(t)dt::::; o.
}Er }ErThis implies u(x, t) = 0 on Er for all r > 0. D
3.5. Strong maximum principle for weak solutions.
For simplicity we shall only state the result below for weak supersolu-
tions. This result is also known as a weak Harnack inequality in the
literature.
Let (Mn,g) be a Riemannian manifold and let 0 EM be such that the
following two properties hold.
(1) For some R, P > 0 and K, > 1, we have for any C^00 function
f : B ( 0, K,r) ---+ JR.,
where r E (0, R], that
(26.124) r If - !avgl^2 dμ::::; Pr^2 r IV' fl^2 dμ,
j B(O,r) j B(O,,,,,r)where !avg ~ Voli(O,r) JB(O,r) f dμ.
(2) There exists D < oo such that for any r E (0, R],
(26.125) VolB(O, 2r)::::; DVolB(O, r).