Eigenvalue optimization in higher dimensions and $p$-harmonic maps

TL;DR

This paper establishes existence of eigenvalue optimization solutions on higher-dimensional manifolds, linking maximizers to p-harmonic maps and employing topological tensor product techniques for analysis.

Contribution

It extends eigenvalue optimization results to higher dimensions and conformal classes, and connects maximizers to p-harmonic maps using novel tensor product methods.

Findings

Existence of eigenvalue maximizers in higher dimensions and conformal classes.

Maximizers are induced by p-harmonic maps into spheres.

Regularity results for maximizers depending on p, with no bubbling for p<m.

Abstract

We prove existence results for optimization problems for the $k$th Laplace eigenvalue on closed Riemannian manifolds of dimension $m \geq 3$, depending on the choice of normalization. One such normalization leads to eigenvalue optimization within a conformal class, for which existence of maximizers was previously known only in dimension two. We also prove that all absolutely continuous maximizers of the normalized eigenvalue functionals are always induced by $p$-harmonic maps into spheres, where $p \in [2,m]$. For $p$ sufficiently close to $m$, the maximizers are always H\"older-continuous, whereas for $p<m$ no bubbling occurs. A key tool in our analysis is the application of techniques from the theory of topological tensor products, which appear to be well suited for studying eigenvalue-related optimization problems.