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 $m\ge 3$. Extending the work of Karpukhin and Stern on the first eigenvalue, we prove that, for every $k\ge 1$, 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 $3\le m\le 6$. We further prove that this dimension bound is optimal: for every $m\ge 7$ and every integer $0\le d\le m-7$, there exists a maximizing harmonic map on the $m$-dimensional round sphere whose singular set has Hausdorff dimension $d$.

Theorem 1.1 strengthened via the addition of Section 5.2; former Section 1.4.1 removed as no longer needed