Maximizing higher eigenvalues in dimensions three and above

TL;DR

This paper investigates the maximization of the $k$-th eigenvalue on higher-dimensional manifolds, extending previous results to dimensions 3 and above, and characterizing the regularity and singularities of the maximizing harmonic maps.

Contribution

It generalizes the characterization of maximizing measures for eigenvalues to dimensions 3 through 6 and describes the singularity structure of maximizers in higher dimensions, establishing bounds on singular set dimensions.

Findings

Maximizers are smooth harmonic maps into spheres for dimensions 3 to 6.

In higher dimensions, maximizers may have singularities with controlled Hausdorff dimension.

The optimal upper bound for singular set dimension is $m-7$.

Abstract

We study the problem of maximizing the $k$-th eigenvalue functional over the class of absolutely continuous measures on a closed Riemannian manifold of dimension $m\geq 3$. For dimensions $3 \leq m \leq 6$, we generalize the work of Karpukhin and Stern on the first eigenvalue, showing that the maximizing measures are realized by smooth harmonic maps into finite-dimensional spheres. For $m \geq 7$, the maximizing measures are again induced by harmonic maps, which may now exhibit singularities. We prove that $m-7$ is the optimal upper bound for the Hausdorff dimension of the singular set. More precisely, for any $m \geq 7$, there exist maximizing harmonic maps on the $m$-dimensional sphere whose singular sets have any prescribed integer dimension up to $m - 7$.