’ll discuss recent progress on the existence theory for harmonic maps, in particular the existence of harmonic maps of optimal regularity from manifolds of dimension n>2 to every non-aspherical closed manifold containing no stable minimal two-spheres. As an application, we’ll see that every manifold carries a canonical family of sphere-valued harmonic maps, which (in dimension<6) stabilize at a solution of a spectral isoperimetric problem generalizing the conformal maximization of  Laplace eigenvalues on surfaces.

Based on joint work with Mikhail Karpukhin.