On the isoperimetric inequality for the first positive Neumann eigenvalue on the sphere

TL;DR

This paper proves that geodesic disks uniquely maximize the first positive Neumann eigenvalue among simply connected domains of fixed area on the sphere.

Contribution

It establishes the isoperimetric inequality for the first positive Neumann eigenvalue on the sphere, identifying geodesic disks as the unique maximizers.

Findings

Geodesic disks are the unique maximizers of the first positive Neumann eigenvalue.

The result applies to all simply connected domains of fixed area on the sphere.

The proof confirms the isoperimetric inequality for this eigenvalue on the sphere.

Abstract

We prove that the geodesic disks are the unique maximisers of the first non-trivial Neumann eigenvalue among all simply connected domains of the sphere $\mathbb S^2$ with fixed area.