Isoperimetric inequality for the third eigenvalue of the Laplace-Beltrami operator on $\mathbb S^2$

TL;DR

This paper establishes an isoperimetric inequality for the third eigenvalue of the Laplace-Beltrami operator on the 2-sphere, extending extremal metric construction techniques to this spectral problem.

Contribution

It introduces a new isoperimetric inequality for the third eigenvalue on the sphere, based on a developed theory of extremal metrics in conformal classes.

Findings

Proves an inequality analogous to Hersch's for the third eigenvalue.

Develops a method to construct extremal metrics on Riemannian surfaces.

Extends spectral geometry techniques to higher eigenvalues.

Abstract

We prove an Hersch's type isoperimetric inequality for the third positive eigenvalue on $\mathbb S^2$. Our method builds on the theory we developped to construct extremal metrics on Riemannian surfaces in conformal classes for any eigenvalue.