Li-Yau gradient estimate for compact manifolds with negative part of   Ricci curvature in the Kato class
  This article shows that if the negative part of Ricci curvature lies in the Kato class, the heat kernel satisfies a Li-Yau type estimate. Additionally, using the resulting heat kernel bound, we show that the obtained heat kernel estimate leads to bounds on the first Betti number only depending on the Kato constant. 
Christian Rose
math.DG