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

Luigi Provenzano; Alessandro Savo

spectral geometry

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

arXiv:2603.04236

summary

The authors prove that among simply connected domains on the unit sphere with a given area, geodesic disks uniquely maximize the first positive Neumann eigenvalue, and they show that this optimality fails for spherical annuli.

Abstract

We prove that geodesic disks are the unique maximizers of the first positive Neumann eigenvalue among all simply connected domains in $S^{2}$ with prescribed area. We then show that geodesic disks are no longer maximisers for the class of spherical annuli.

The paper is an improvement of the previous version

Topics & keywords

#isoperimetric inequality #neumann eigenvalue #spherical domains #geodesic disks #spectral optimizationfirst positive Neumann eigenvaluegeodesic diskS^2spectral isoperimetric problemspherical annulus