Download this article
Download this article For screen
For printing
Recent Issues

Volume 17
Issue 4, 1127–1500
Issue 3, 757–1126
Issue 2, 379–756
Issue 1, 1–377

Volume 16, 10 issues

Volume 15, 8 issues

Volume 14, 8 issues

Volume 13, 8 issues

Volume 12, 8 issues

Volume 11, 8 issues

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the journal
Ethics and policies
Peer-review process
Submission guidelines
Submission form
Editorial board
Editors' interests
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Author index
To appear
Other MSP journals
Eigenvalue estimates for Kato-type Ricci curvature conditions

Christian Rose and Guofang Wei

Vol. 15 (2022), No. 7, 1703–1724

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.

Kato condition, variable Ricci curvature, eigenvalue estimate, heat equation
Mathematical Subject Classification 2010
Primary: 53C21, 58J35, 58J50
Received: 16 March 2020
Revised: 23 September 2020
Accepted: 16 March 2021
Published: 5 December 2022
Christian Rose
Institut für Mathematik
Universität Potsdam
Guofang Wei
Department of Mathematics
University of California
Santa Barbara, CA
United States