Talk page
Title:
Harmonic maps and extremal Schrodinger operators on higher-dimensional manifolds
Speaker:
Abstract:
I'll discuss recent progress on the existence theory for harmonic maps from arbitrary Riemannian manifolds of dimension >2 to a large class of target manifolds, including 3-manifolds of positive Ricci curvature and manifolds of dimension > 3 with positive isotropic curvature. As a special case, every Riemannian manifold (M^n,g) of dimension n>2 admits a family of nontrivial stationary harmonic maps u_k to the standard spheres S^k for k>2, smooth away from a singular set of dimension at most n-7 for k sufficiently large. I'll explain how these maps give rise to solutions of an optimization problem for Schrodinger operators related to work of Grigor'yan-Netrusov-Yau and Grigor'yan-Nadirashvili-Sire, generalizing to higher dimensions the maximization of Laplace eigenvalues on surfaces of fixed conformal type. (Based on joint work with Mikhail Karpukhin.)
Link:
Workshop: