Maximization of the second conformal eigenvalue of spheres

TL;DR

This paper derives an upper bound for the second eigenvalue of spheres within their conformal class, applicable across all dimensions and becoming nearly optimal as the dimension grows.

Contribution

It provides a universal upper bound on the second conformal eigenvalue of spheres, extending previous results to all dimensions with asymptotic sharpness.

Findings

Upper bound on second eigenvalue established

Bound is asymptotically sharp in high dimensions

Applicable to all n-dimensional spheres

Abstract

We establish in this paper an upper bound on the second eigenvalue of n-dimensional spheres in the conformal class of the round sphere. This upper bound holds in all dimensions and is asymptotically sharp as the dimension increases.