We will discuss the existence of metrics on a closed Riemannian manifold of dimensionm ≥ 3 that maximize the kth Laplace eigenvalue within a conformal class. Previously,such existence results were known only in dimension two. More generally, we consider afamily of eigenvalue optimization problems parametrized by the choice of normalization.
The critical points of the resulting normalized eigenvalue functionals are related top-harmonic maps into spheres, where 2 ≤ p ≤ m, with the case p = m correspondingto optimization within a conformal class. A key tool in the analysis is the use oftechniques from the theory of topological tensor products, which appear to be well suitedfor studying eigenvalue optimization problems.
