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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 Lp-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
Milestones
Received: 16 March 2020
Revised: 23 September 2020
Accepted: 16 March 2021
Published: 5 December 2022
Authors
Christian Rose
Institut für Mathematik
Universität Potsdam
Potsdam
Germany
Guofang Wei
Department of Mathematics
University of California
Santa Barbara, CA
United States