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

Abstract

We prove existence results for optimization problems for the $k$th Laplace eigenvalue on closed Riemannian manifolds of dimension $m\ge 3$, 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 Hölder-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.

48 pages; corrected proof of Proposition 4.1; misc typo fixes