Skip to main content
SpringerLink
Log in

Plateau’s Problem and Riemann’s Mapping Theorem

  • Published:
Milan Journal of Mathematics Aims and scope Submit manuscript

Abstract

First the connection between conformal mappings and Plateau’s problem is pointed out. Then the relation between minimizers of area and energy under Plateau boundary conditions is discussed (joint work with F. Sauvigny). Finally, generalizations of the mapping theorems of Riemann and Koebe for Riemannian metrics are presented (joint work with H. von der Mosel).

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

Bibliography

  1. Courant R.: Dirichlet’s principle, conformal mapping, and minimal surfaces. Interscience Publ., New York (1950)

    MATH  Google Scholar 

  2. Dierkes, U., Hildebrandt, S., Sauvigny, F.: Minimal surfaces. Grundlehren der math. Wiss. Vol. 339, Springer, Heidelberg 2010.

  3. Dierkes, U., Hildebrandt, S., Tromba, A.: Regularity of minimal surfaces. Grundlehren der math. Wiss. Vol. 340, Springer, Heidelberg 2010.

  4. Dierkes, U., Hildebrandt, S., Tromba, A.: Global analysis of minimal surfaces. Grundlehren der math. Wiss. Vol. 341, Springer, Heidelberg 2010.

  5. Hildebrandt S., von der Mosel H.: On Lichtenstein’s theorem about globally conformal mappings. Calc. Var. 23, 415–424 (2005)

    Article  MATH  Google Scholar 

  6. Hildebrandt S., von der Mosel H.: Conformal mapping of multiply connected Riemann domains by a variational approach. Adv. Calc. Var. 2, 137–183 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hildebrandt S., Sauvigny F.: Relative minimizers of energy are relative minimizers of area. Calc. Var. 37, 475–483 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Jost J.: Conformal mappings and the Plateau-Douglas problem in Riemannian manifolds. J. reine u. angewandte Math. 359, 37–54 (1984)

    MathSciNet  Google Scholar 

  9. Jost J.: Two-dimensional geometric variational problems. Wiley, New York (1990)

    Google Scholar 

  10. Lichtenstein, L.: Zur Theorie der konformen Abbildung: Konforme Abbildung nichtanalytischer singularitätenfreier Flächenstücke auf ebene Gebiete. Bull. Acad. Sci. Cracovie, Cl. Sci. Math. Nat. Ser. A, 192–217 (1916).

  11. Morrey, C. B.: Multiple integrals in the calculus of variations. Grundlehren der math. Wiss. Vol. 130, Springer, Berlin 1966.

  12. Sauvigny F.: Introduction of isothermal parameters into a Riemannian metric by the continuity method. Analysis 19, 235–243 (1999)

    MathSciNet  MATH  Google Scholar 

  13. Sauvigny, F.: Partial differential equations 1 & 2. Universitext, Springer, Berlin 2003/2004.

  14. Vekua, I. N.: Verallgemeinerte analytische Funktionen. Akademie-Verlag Berlin, 1963.

  15. Warschawski, S.: \"Uber einen Satz von O. D. Kellogg. Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. 73–86 (1932).

Download references

Author information

Authors and Affiliations

  1. Mathematisches Institut, Universität Bonn, Endenicher Allee 60, D-53115, Bonn, Germany

    Stefan Hildebrandt

Authors
  1. Stefan Hildebrandt
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stefan Hildebrandt.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hildebrandt, S. Plateau’s Problem and Riemann’s Mapping Theorem. Milan J. Math. 79, 67–79 (2011). https://doi.org/10.1007/s00032-011-0142-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00032-011-0142-y

Keywords

Search

Navigation