Eigenvalue estimates for Kato-type Ricci curvature conditions

Rose, Christian; Wei, Guofang

doi:10.2140/apde.2022.15.1703

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Eigenvalue estimates for Kato-type Ricci curvature conditions

Christian Rose and Guofang Wei

Vol. 15 (2022), No. 7, 1703–1724

DOI: 10.2140/apde.2022.15.1703

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

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

Vol. 15, No. 7, 2022