Harmonic diffeomorphisms between domains in the Euclidean 2-sphere

Abstract

We study the existence or non-existence of harmonic diffeomorphisms between certain domains in the Euclidean $2$-sphere. In particular, we show the existence of harmonic diffeomorphisms from circular domains in the complex plane onto finitely punctured spheres, with at least two punctures. This result follows from a general existence theorem for maximal graphs in the Lorentzian product $\mathbb{M}\times {\mathbb{R}}_1$, where $\mathbb{M}$ is an arbitrary $\mathfrak{n}$-dimensional compact Riemannian manifold, $\mathfrak{n}\ge 2$. In contrast, we show that there is no harmonic diffeomorphism from the unit complex disc onto the once-punctured sphere and no harmonic diffeomeorphisms from finitely punctured spheres onto circular domains in the Euclidean $2$-sphere.