  We prove that optimal lower eigenvalue estimates of Zhong-Yang type as well
as a Cheng-type upper bound for the first eigenvalue hold on closed manifolds
assuming only a Kato condition on the negative part of the Ricci curvature.
This generalizes all earlier results on $L^p$-curvature assumptions. Moreover,
we introduce the Kato condition on compact manifolds with boundary with respect
to the Neumann Laplacian, leading to Harnack estimates for the Neumann heat
kernel and lower bounds for all Neumann eigenvalues, what provides a first
insight in handling variable Ricci curvature assumptions in this case.

Differential Geometry;Analysis of PDEs;Spectral Theory