Maximization of the second Laplacian Eigenvalue on the Sphere

TL;DR

This paper proves a sharp isoperimetric inequality for the second Laplacian eigenvalue on spheres, confirming a conjecture for higher dimensions and providing simpler proofs for the 2-sphere case.

Contribution

It establishes the conjectured upper bound for the second eigenvalue on higher-dimensional spheres using new methods based on hyperbolic center of mass.

Findings

Maximal second eigenvalue occurs when the sphere degenerates into two disjoint spheres for S^2.

Confirmed the conjecture for higher-dimensional spheres.

Provided a simplified proof for the case of S^2.

Abstract

We prove a sharp isoperimetric inequality for the second nonzero eigenvalue of the Laplacian on $S^m$. For $S^{2}$, the second nonzero eigenvalue becomes maximal as the surface degenerates to two disjoint spheres, by a result of Nadirashvili for which Petrides later gave another proof. For higher dimensional spheres, the analogous upper bound was conjectured by Girouard, Nadirashvili and Polterovich. Our method to confirm the conjecture builds on Petrides' work and recent developments on the hyperbolic center of mass and provides also a simpler proof for $S^2$.