Harmonic maps from surfaces of arbitrary genus into spheres

Abstract

We relate the existence problem of harmonic maps into S^2 to the convex geometry of S^2. On one hand, this allows us to construct new examples of harmonic maps of degree 0 from compact surfaces of arbitrary genus into S^2. On the other hand, we produce new examples of regions that do not contain closed geodesics (that is, harmonic maps from S^1) but do contain images of harmonic maps from other domains. These regions can therefore not support a strictly convex functions. Our construction uses M. Struwe’s heat flow approach for the existence of harmonic maps from surfaces.