Harmonic diffeomorphisms between domains in the Euclidean 2-sphere

  • Antonio Alarcón

    Universidad de Granada, Spain
  • Rabah Souam

    Institut Mathématiques de Jussieu, Paris, France


We study the existence or non-existence of harmonic diffeomorphisms between certain domains in the Euclidean -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 , where is an arbitrary -dimensional compact Riemannian manifold, . 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 -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