On the eigenvalues of the Laplacian on ellipsoids obtained as perturbation of unit sphere

TL;DR

This paper investigates how the eigenvalues of the Laplacian on ellipsoids, viewed as perturbations of the unit sphere, compare under certain curvature conditions, extending classical geometric spectral results.

Contribution

It provides a comparison of Laplacian eigenvalues on perturbed ellipsoids with those on the sphere under Gaussian curvature constraints, linking to Lichnerowicz's theorem.

Findings

Eigenvalues of ellipsoids are compared to those of the sphere.

A Gaussian curvature condition influences the eigenvalue comparison.

Results extend classical spectral geometry theorems.

Abstract

We study the eigenvalues of the Laplacian on ellipsoids that are obtained as perturbations of the standard Euclidean unit sphere in dimension two. A comparison of these eigenvalues with those of the standard Euclidean unit sphere is obtained under a Gaussian curvature condition, in line with the Lichnerowicz theorem on the first positive eigenvalue on a compact Riemannian manifold.