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

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 Hölder-continuous, whereas for $p