Maximizing higher eigenvalues in dimensions three and above

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 $m\ge 3$. For dimensions $3\le m\le 6$, 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 $m\ge 7$, 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 $m\ge 7$, there exist maximizing harmonic maps on the $m$-dimensional sphere whose singular sets have any prescribed integer dimension up to $m-7$.