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Harmonic mappings and conformal minimal immersions of Riemann surfaces into \({\mathbb {R}^{\rm N}}\)

  • Antonio Alarcón
  • Isabel Fernández
  • Francisco J. López
Article

Abstract

We prove that for any open Riemann surface \({\mathcal{N}}\), natural number N ≥ 3, non-constant harmonic map \({h:\mathcal{N} \to \mathbb{R}}\) N−2 and holomorphic 2-form \({\mathfrak{H}}\) on \({\mathcal{N}}\) , there exists a weakly complete harmonic map \({X=(X_j)_{j=1,\ldots,{\sc N}}:\mathcal{N} \to \mathbb{R}^{\sc N}}\) with Hopf differential \({\mathfrak{H}}\) and \({(X_j)_{j=3,\ldots,{\sc N}}=h.}\) In particular, there exists a complete conformal minimal immersion \({Y=(Y_j)_{j=1,\ldots,{\sc N}}:\mathcal{N} \to \mathbb{R}^{\sc N}}\) such that \({(Y_j)_{j=3,\ldots,{\sc N}}=h}\) . As some consequences of these results (1) there exist complete full non-decomposable minimal surfaces with arbitrary conformal structure and whose generalized Gauss map is non-degenerate and fails to intersect N hyperplanes of \({\mathbb{CP}^{{\sc N}-1}}\) in general position. (2) There exist complete non-proper embedded minimal surfaces in \({\mathbb{R}^{\sc N},}\) \({\forall\,{\sc N} >3 .}\)

Mathematics Subject Classification

53C43 53C42 30F15 49Q05 

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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Antonio Alarcón
    • 1
  • Isabel Fernández
    • 2
  • Francisco J. López
    • 1
  1. 1.Departamento de Geometría y TopologíaUniversidad de GranadaGranadaSpain
  2. 2.Departamento de Matemática Aplicada IUniversidad de SevillaSevillaSpain

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