  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 example 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 function. Our construction builds upon an example of W. Kendall, and
uses M. Struwe's heat flow approach for the existence of harmonic maps from
surfaces.
