Geometry and Topology Seminar

In this talk, I will discuss a correspondence between extremal problems from spectral geometry and geometric object such as n-harmonic maps. On a Riemannian manifold M, one can study the eigenvalues of the Laplacian or of the Steklov problem as functionals of the metric. Of particular interest are their critical points, which we call extremal metrics. If is known that for the Laplace eigenvalues, extremal metrics correspond to minimal immersions to a sphere; while for the Steklov eigenvalues on a surface, extremal metrics give rise to free-boundary minimal surface in a ball. I will discuss higher dimensional generalization of this Steklov result and how one obtains n-harmonic maps when considering extremal metrics in a conformal class. This is joint work with Mikhail Karpukhin.