Conformal and extrinsic upper bounds for the harmonic mean of Neumann and Steklov eigenvalues

TL;DR

This paper establishes new conformal and extrinsic upper bounds for the harmonic mean of the first nonzero Neumann and Steklov eigenvalues on manifolds, extending previous results and involving conformal volumes.

Contribution

It introduces novel upper bounds for eigenvalue means using conformal and extrinsic geometric quantities, generalizing earlier findings.

Findings

Derived upper bounds involving conformal volume and relative conformal volume.

Provided an optimal sharp extrinsic upper bound for closed submanifolds in space forms.

Extended previous results to the harmonic mean of multiple eigenvalues.

Abstract

Let $M$ be an $m$-dimensional compact Riemannian manifold with boundary. We obtain the upper bound of the harmonic mean of the first $m$ nonzero Neumann eigenvalues and Steklov eigenvalues involving the conformal volume and relative conformal volume, respectively. We also give an optimal sharp extrinsic upper bound for closed submanifolds in space forms. These extend the previous related results for the first nonzero eigenvalues.