Heat kernel upper bound on Riemannian manifolds with locally uniform   Ricci curvature integral bounds
  This article shows that under locally uniformly integral bounds of the negative part of Ricci curvature the heat kernel admits a Gaussian upper bound for small times. This provides general assumptions on the geometry of a manifold such that certain function spaces are in the Kato class. Additionally, the results imply bounds on the first Betti number. 
Christian Rose
math.DG