JournalscmhVol. 89, No. 1pp. 255–271

Harmonic diffeomorphisms between domains in the Euclidean 2-sphere

  • Antonio Alarcón

    Universidad de Granada, Spain
  • Rabah Souam

    Institut Mathématiques de Jussieu, Paris, France
Harmonic  diffeomorphisms between domains in the Euclidean 2-sphere cover
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Abstract

We study the existence or non-existence of harmonic diffeomorphisms between certain domains in the Euclidean 22-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 M×R1\mathbb{M}\times\mathbb{R}_1, where M\mathbb{M} is an arbitrary n\mathfrak{n}-dimensional compact Riemannian manifold, n2\mathfrak{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 22-sphere.

Cite this article

Antonio Alarcón, Rabah Souam, Harmonic diffeomorphisms between domains in the Euclidean 2-sphere. Comment. Math. Helv. 89 (2014), no. 1, pp. 255–271

DOI 10.4171/CMH/318