 
Summary: A Heat Kernel Lower Bound for Integral Ricci Curvature
Xianzhe Dai y Guofang Wei z
Abstract
In this note we give a heat kernel lower bound in term of integral Ricci curvature, extending
CheegerYau's estimate.
1 Introduction
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 [2, 7, 1]). The heat kernel upper bound has been
extended to integral Ricci curvature by Gallot in [4]. Here we extend CheegerYau's lower bound
[2] to integral Ricci curvature.
Our notation for the integral curvature bounds on a Riemannian manifold (M; g) is as follows.
For each x 2 M let r (x) denote the smallest eigenvalue for the Ricci tensor Ric : T x M ! T x M; and
for any xed number dene
(x) = jmin f0; r (x) (n 1)gj :
Then set
k (p; ; R) = sup
x2M
Z
B(x;R)
p
