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Eigenvalue estimates for Kato-type Ricci curvature conditions

### Christian Rose and Guofang Wei

Vol. 15 (2022), No. 7, 1703–1724
##### Abstract

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

##### Keywords
Kato condition, variable Ricci curvature, eigenvalue estimate, heat equation
##### Mathematical Subject Classification 2010
Primary: 53C21, 58J35, 58J50