Maximization of higher order eigenvalues and applications

TL;DR

This paper studies how to maximize higher order eigenvalues within conformal classes on smooth compact surfaces, revealing bubbling phenomena that differ from the behavior of the first eigenvalue.

Contribution

It extends previous work by analyzing higher order eigenvalues, demonstrating the occurrence of bubbling phenomena in the maximization process.

Findings

Bubbling phenomena occur in the maximization of higher order eigenvalues.

The behavior differs from the case of the first eigenvalue.

Provides new insights into spectral geometry on Riemannian surfaces.

Abstract

The present paper is a follow up of our paper \cite{nS}. We investigate here the maximization of higher order eigenvalues in a conformal class on a smooth compact boundaryless Riemannian surface. Contrary to the case of the first nontrivial eigenvalue as shown in \cite{nS}, bubbling phenomena appear.