# An isoperimetric inequality for Laplace eigenvalues on the sphere

## TL;DR

This paper proves a sharp isoperimetric inequality for Laplace eigenvalues on the sphere, showing maximizers are limits of metrics converging to unions of identical spheres, confirming a long-standing conjecture.

## Contribution

It establishes the maximization of the k-th eigenvalue by a sequence of metrics approaching a union of k touching spheres, extending previous results for small k.

## Key findings

- Maximizers are limits of metrics converging to unions of k spheres.
- The supremum is not attained by smooth metrics for k≥2.
- A new bound on harmonic map degree is introduced.

## Abstract

We show that for any positive integer k, the k-th nonzero eigenvalue of the Laplace-Beltrami operator on the two-dimensional sphere endowed with a Riemannian metric of unit area, is maximized in the limit by a sequence of metrics converging to a union of k touching identical round spheres. This proves a conjecture posed by the second author in 2002 and yields a sharp isoperimetric inequality for all nonzero eigenvalues of the Laplacian on a sphere. Earlier, the result was known only for k=1 (J. Hersch, 1970), k=2 (N. Nadirashvili, 2002; R. Petrides, 2014) and k=3 (N. Nadirashvili and Y. Sire, 2017). In particular, we argue that for any k>=2, the supremum of the k-th nonzero eigenvalue on a sphere of unit area is not attained in the class of Riemannian metrics which are smooth outsitde a finite set of conical singularities. The proof uses certain properties of harmonic maps between spheres, the key new ingredient being a bound on the harmonic degree of a harmonic map into a sphere obtained by N. Ejiri.