Title: Maximizing Laplace eigenvalues with density in higher dimensions

Abstract: We will discuss the problem of maximizing the k-th Laplace eigenvalue with density on a closed Riemannian manifold of dimension m ≥ 2. 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.