Mathematische Zeitschrift

, Volume 283, Issue 1–2, pp 1–24 | Cite as

Embedded minimal surfaces in \({\mathbb {R}}^n\)

  • Antonio Alarcón
  • Franc Forstnerič
  • Francisco J. López
Article

Abstract

In this paper, we prove that every conformal minimal immersion of an open Riemann surface into \({\mathbb {R}}^n\) for \(n\ge 5\) can be approximated uniformly on compacts by conformal minimal embeddings (see Theorem 1.1). Furthermore, we show that every open Riemann surface carries a proper conformal minimal embedding into \({\mathbb {R}}^5\) (see Theorem 1.2). One of our main tools is a Mergelyan approximation theorem for conformal minimal immersions to \({\mathbb {R}}^n\) for any \(n\ge 3\) which is also proved in the paper (see Theorem 5.3).

Keywords

Riemann surfaces Minimal surfaces Conformal minimal embeddings 

Mathematics Subject Classfication

53A10 32B15 32E30 32H02 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Antonio Alarcón
    • 1
  • Franc Forstnerič
    • 2
    • 3
  • Francisco J. López
    • 4
  1. 1.Departamento de Geometría y Topología e Instituto de Matemáticas (IEMath-GR)Universidad de GranadaGranadaSpain
  2. 2.Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia
  3. 3.Institute of Mathematics, Physics and MechanicsLjubljanaSlovenia
  4. 4.Departamento de Geometría y TopologíaUniversidad de GranadaGranadaSpain

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