Harmonic diffeomorphisms between domains in the Euclidean 2-sphere

TL;DR

This paper investigates the existence of harmonic diffeomorphisms between domains in the Euclidean 2-sphere, establishing conditions under which such mappings exist or do not, with implications for maximal graphs in Lorentzian products.

Contribution

It provides a general existence theorem for harmonic diffeomorphisms from circular domains to finitely punctured spheres and shows non-existence results in specific cases.

Findings

Harmonic diffeomorphisms exist from circular domains to finitely punctured spheres with at least two punctures.

No harmonic diffeomorphism from the unit disc to the once punctured sphere exists.

Certain harmonic diffeomorphisms between punctured spheres and circular domains do not exist.

Abstract

We study the existence or not of harmonic diffeomorphisms between certain domains in the Euclidean 2-sphere. In particular, we show 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 $M\times\mathbb{R}_1,$ where $M$ is an arbitrary $n$-dimensional compact Riemannian manifold, $n\geq 2.$ 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.