- RIEMANNIAN GEOMETRY 453
1.12. Li-Yau differential Harnack estimate. Let u : Mn x [O, oo) --+
~ be a positive solution to the heat equation ~~ = flu on a complete Rie-
mannian manifold (Mn,g). Define f by(A.11) u ~ (47rt)-nl^2 e-f,so that 8ft = b..f - [\7 f[^2 - ~-
THEOREM A.10 (Li-Yau differential Harnack estimate). If (Mn, g) has
nonnegative Ricci curvature, then(A.12) b..f - -n = -of + IV f I^2 < 0.
2t at -
Integrating (A.12) yields the following sharp version of the classical Har-
nack estimate:(A.13)for all xi, x2 E M and t2 > ti. If u = H is a fundamental solution centered
at a point x EM, then taking ti--+ 0 implies the Cheeger-Yau estimate:(A.14) f ( y, t) < - d ( x' 4t y)^2
In terms of u, the positive solution to the heat equation, on a complete
Riemannian manifolds with nonnegative Ricci curvature, we have
(^0 2) n
at logu - [Vlogu[ = tllogu 2: - 2 t
and
u(x2,t2) (t2)-n/
2
{ d(xi,x2)
2
---> - exp }.
u (xi, ti) - ti 4 (t2 - ti)
For a fundamental solution u = H,
H (y, t) ~ ( 47rt)-n/2 exp {-d (~ty)'}.
1.13. Calabi's trick. In this subsection we give an example of Calabi's
trick which is useful in the study of heat-type equations and analytic aspects
of the Ricci fl.ow. In particular, a slight modification of the discussion below
applies to the proof of the local first derivative of curvature estimate for the
Ricci fl.ow (see Theorem A.30).
First, let us recall some facts about the distance function. Let (Mn, g) be
a Riemannian manifold. Given p EM, the distance function r (x) ~ d (x,p)
is Lipschitz on M with Lipschitz constant 1. Let Cut (p) denote the cut
locus of p and let
Gp~ {VE TpM: d (p, expP (V)) = IVI},