Eigenvalue inequalities on Riemannian manifolds with a lower Ricci curvature bound

TL;DR

This paper extends classical eigenvalue inequalities to Riemannian manifolds with boundary, providing new bounds for Dirichlet and Neumann problems and discussing eigenvalue multiplicity and open questions.

Contribution

It introduces versions of classical eigenvalue inequalities for boundary value problems on manifolds with boundary, expanding their applicability.

Findings

Eigenvalue bounds for Dirichlet and Neumann problems established

Eigenvalue multiplicity bounds discussed

Open problems in eigenvalue theory highlighted

Abstract

We revisit classical eigenvalue inequalities due to Buser, Cheng, and Gromov on closed Riemannian manifolds, and prove the versions of these results for the Dirichlet and Neumann boundary value problems. Eigenvalue multiplicity bounds and related open problems are also discussed.