Geometry & Topology

Eigenvalue optimization for the Laplacian has a remarkable connection with harmonic map theory. We will discuss this connection for the problem of maximizing Laplace eigenvalues within a fixed conformal class on a closed Riemannian manifold of dimension $m \geq 3$. More generally, we consider a family of weighted eigenvalue optimization problems depending on a normalization parameter. Their critical points are related to $p$-harmonic maps into spheres, where $2 \leq p \leq m$, with $p = m$ corresponding to the conformally invariant problem. We will discuss the existence of maximizing metrics and weights and describe some examples for which these optimization problems can be solved explicitly.