Skip to main content
SpringerLink
Log in

Limit sets for complete minimal immersions

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

In this paper we study the behaviour of the limit set of complete proper compact minimal immersions in a domain \(G \subset {\mathbb{R}}^3\) with the boundary \(\partial G \subset C^2.\) We prove that the second fundamental form of the surface ∂G is nonnegatively defined at every point of the limit set of such immersions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alarcón, A., Ferrer, L., Martín, F.: Density theorems for complete minimal surfaces in \({\mathbb{R}}^3\) . Geom. Funct. Anal. (to appear)

  2. Bourgain J. (1993). On the radial variation of bounded analytic functions on the disc. Duke Math. J. 69(3): 671–682

    Article  MATH  MathSciNet  Google Scholar 

  3. López F.J., Martín F. and Morales S. (2002). Adding handles to Nadirashvili’s surfaces. J. Differ. Geom. 60: 155–175

    MATH  Google Scholar 

  4. Martín F., Meeks W.H. and Nadirashvili N. (2007). Bounded domains which are universal for minimal surfaces. Am. J. Math. 129(2): 455–461

    Article  MATH  Google Scholar 

  5. Martín F. and Morales S. (2004). On the asymptotic behavior of a complete bounded minimal surface in \({\mathbb{R}}^3\) T. Am. Math. Soc. 356(10): 3985–3994

    Article  MATH  Google Scholar 

  6. Martín F. and Morales S. (2005). Complete proper minimal surfaces in convex bodies of \({\mathbb{R}}^3\) Duke Math. J. 128: 559–593

    Article  MATH  MathSciNet  Google Scholar 

  7. Martín F. and Morales S. (2006). Complete proper minimal surfaces in convex bodies of \({\mathbb{R}}^3\) (II): the behavior of the limit setComment. Math. Helv. 81: 699–725

    Article  MATH  MathSciNet  Google Scholar 

  8. Martín F. and Nadirashvili N. (2007). A Jordan curve spanned by a complete minimal surface. Arch. Ration. Mech. An. 184(2): 285–301

    Article  MATH  Google Scholar 

  9. Nadirashvili N. (1996). Hadamard’s and Calabi-Yau’s conjectures on negatively curved and minimal surfaces. Invent. Math. 126: 457–465

    Article  MATH  MathSciNet  Google Scholar 

  10. Nadirashvili N. (2001). An application of potential analysis to minimal surfaces. Mosc. Math. J. 1(4): 601–604

    MATH  MathSciNet  Google Scholar 

  11. Osserman R. (1971). The convex hull property of immersed manifolds. J. Differ. Geom. 6: 267–270

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Geometría y Topología, Universidad de Granada, 18071, Granada, Spain

    Antonio Alarcón

  2. CNRS, LATP, CMI, 39, rue Joliot-Curie, 13453, Marseille Cedex 13, France

    Nikolai Nadirashvili

Authors
  1. Antonio Alarcón
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nikolai Nadirashvili
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Antonio Alarcón.

Additional information

A. Alarcón’s research is partially supported by MEC-FEDER Grant no. MTM2004-00160.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Alarcón, A., Nadirashvili, N. Limit sets for complete minimal immersions. Math. Z. 258, 107–113 (2008). https://doi.org/10.1007/s00209-007-0161-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-007-0161-0

Keywords

Mathematics Subject Classification (2000)

Search

Navigation