Maximizing higher eigenvalues in dimensions three and above

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 $m\geq 3$. Extending the work of Karpukhin and Stern on the first eigenvalue, we prove that, for every $k\geq 1$, the supremum is attained by a measure induced by a harmonic map into a finite-dimensional sphere. The map is smooth outside a closed singular set of Hausdorff dimension at most $m-7$, and is therefore smooth when $3 \leq m \leq 6$. We further prove that this dimension bound is optimal: for every $m \geq 7$ and every integer $0\leq d \leq m-7$, there exists a maximizing harmonic map on the $m$-dimensional round sphere whose singular set has Hausdorff dimension $d$.