Laplace and Steklov extremal metrics via $n$-harmonic maps

TL;DR

This paper introduces a unified framework using $n$-harmonic maps to describe extremal metrics for Laplace and Steklov eigenvalues on manifolds of any dimension, revealing new features and geometric interpretations.

Contribution

It extends classical results linking extremal metrics to harmonic maps to higher dimensions and uncovers new properties of Steklov eigenvalues, including a natural normalization and geometric interpretation.

Findings

Unique normalization involving boundary and volume in higher dimensions

Critical points have a geometric interpretation with densities

Construction of free boundary harmonic annuli in the 3D ball

Abstract

We present a unified description of extremal metrics for the Laplace and Steklov eigenvalues on manifolds of arbitrary dimension using the notion of $n$-harmonic maps. Our approach extends the well-known results linking extremal metrics for eigenvalues on surfaces with minimal immersions and harmonic maps. In the process, we uncover two previously unknown features of the Steklov eigenvalues. First, we show that in higher dimensions there is a unique normalization involving both the volume of the boundary and of the manifold itself, which leads to meaningful extremal eigenvalue problems. Second, we observe that the critical points of the eigenvalue functionals in a fixed conformal class have a natural geometric interpretation provided one considers the Steklov problem with a density. As an example, we construct a family of free boundary harmonic annuli in the three-dimensional ball and conjecture that they correspond to metrics maximizing the first Steklov eigenvalue in their respective conformal classes.