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
-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, which
provides a first insight in handling variable Ricci curvature assumptions in this
case.
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