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


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).


Riemann surfaces Minimal surfaces Conformal minimal embeddings 

Mathematics Subject Classfication

53A10 32B15 32E30 32H02 



A. Alarcón is supported by the Ramón y Cajal program of the Spanish Ministry of Economy and Competitiveness. A. Alarcón and F. J. López are partially supported by the MINECO/FEDER Grants MTM2011-22547 and MTM2014-52368-P, Spain. F. Forstnerič is partially supported by the research program P1-0291 and the Grant J1-5432 from ARRS, Republic of Slovenia. Part of this work was made when F. Forstnerič visited the institute IEMath-Granada with support by the GENIL-SSV 2014 program. We wish to thank an anonymous referee for the remarks which lead to improved presentation.


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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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