Lower bounds for the sum of the reciprocals of eigenvalues of bounded domains in $\mathbb{R}^n$, spheres, and closed orientable surfaces

TL;DR

This paper establishes lower bounds for the sum of reciprocals of Laplacian eigenvalues across various geometries, extending known results and connecting to conjectures involving conformal volume.

Contribution

It provides new lower bounds for eigenvalue sums in bounded domains, spheres, and surfaces, extending and strengthening existing results and conjectures.

Findings

Lower bounds for eigenvalue reciprocals in bounded domains.

Extension of eigenvalue bounds to spheres and surfaces.

Strengthening of conjectures involving conformal volume.

Abstract

We establish lower bounds for the sum of the reciprocals of eigenvalues of the Laplacian. For bounded domains, our result extends the upper bound provided by Bucur and Henrot on the second Neumann eigenvalue and is related to a result by Wang and Xia, which connects to a conjecture of Ashbaugh and Benguria. For spheres and surfaces, we extend known results on the first and second eigenvalues, and strengthen an analogous conjecture involving the conformal volume of Li and Yau.