 
Summary: Michigan Math. J. 52 (2004)
A Heat Kernel Lower Bound
for Integral Ricci Curvature
Xianzhe Dai & Guofang Wei
1. Introduction
The heat kernel is one of the most fundamental quantities in geometry. It can be
estimated both from above and below in terms of Ricci curvature (see [1; 2; 7]).
The heat kernel upper bound has been extended to integral Ricci curvature by Gal
lot in [4]. Here we extend Cheeger and Yau's [2] lower bound to integral Ricci
curvature.
Our notation for the integral curvature bounds on a Riemannian manifold (M, g)
is as follows. For each x M let r(x) denote the smallest eigenvalue for the Ricci
tensor Ric: Tx M Tx M, and for any fixed number define
(x) = min{0, r(x)  (n  1)}.
Then set
k(p, ,R) = sup
xM B(x,R)
p
1/p
,
