Harmonic maps from surfaces of arbitrary genus into spheres

Abstract

We relate the existence problem of harmonic maps into S2 to the convex geometry of S2. On one hand, this allows us to construct new examples of harmonic maps of degree 0 from compact surfaces of arbitrary genus into S2. On the other hand, we produce new examples of regions that do not contain closed geodesics (that is, harmonic maps from S1) 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.