Harmonic diffeomorphisms between domains in the Euclidean 2-sphere

Details Harmonic diffeomorphisms between domains in the Euclidean 2-sphere Harmonic diffeomorphisms between domains in the Euclidean 2-sphere

2011-08-09

arxiv.org/abs/1108.1960v2

Mathematics

Differential Geometry

13 pages, Main Theorem improved

Scientific paper

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.

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