We will discuss the problem of maximizing the k-th Laplace eigenvalue with density on a closed Riemannian manifold of dimension m ≥ 3. The Euler–Lagrange equation identifies critical densities with the energy densities of harmonic maps into spheres, linking spectral optimization to harmonic-map theory. Unlike the case m = 2, where a priori multiplicity bounds yield existence and regularity, higher dimensions allow unbounded multiplicities.

In the talk, we will present techniques from topological tensor products that handle this setting and prove the existence of maximizing densities for all m ≥ 3. For regularity, optimizers are smooth away from a singular set; when m ≥ 7, this set can have any prescribed integer dimension up to m − 7, as shown by examples on the m-sphere. These techniques have potential for other eigenvalue-optimization problems in higher dimensions where unbounded multiplicities arise.