Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

On immersions of constant mean curvature: Compactness results and finiteness theorems for Plateau's problem

  • 80 Accesses

  • 12 Citations


Assuming stability and integral conditions we show that a sequence of immersed surfaces of constant mean curvatureH converges to an immersedH-surface. The latter theorem depends on an oscillation estimate forH-surfaces based on an isoperimetric inequality. These compactness results are utilized to prove that certain Jordan curvesΓ only bound finitely many stable and unstable, immersed, smallH-surfaces.

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


  1. [Ba]

    Bandle, C.: Konstruktion isoperimetrischer Ungleichungen der mathematischen Physik aus solchen der Geometrie. Comm. Math. Helv.46, 182–213 (1971).

  2. [B 1]

    Barbosa, J. L., &Do Carmo, M.: On the size of a stable minimal surface in\(\mathbb{R}^3 \) 3. Am. J. of Math.98, No. 2, 515–528 (1976).

  3. [B 2]

    Barbosa, J. L., &Do Carmo, M.: A proof of a general isoperimetric inequality for surfaces. Math. Z.162, 245–261 (1978).

  4. [B 3]

    Barbosa, J. L., &Do Carmo, M.: Stability of Minimal surfaces and Eigenvalues of the Laplacian. Math. Z.173, 13–28 (1980).

  5. [Be 1]

    Beeson, M.: Some Results on Finiteness in Plateau's Problem. Part I. Math. Z.175, 103–123 (1980).

  6. [Be 2]

    Beeson, M.: The 6π-theorem about minimal surfaces. Pacific J. of Math.117, 17–25 (1985).

  7. [Bö]

    Böhme, R., &Tomi, F.: Zur Struktur der Lösungsmenge des Plateauproblems. Math. Z.133, 1–29 (1973).

  8. [D]

    Dziuk, G.: Das Verhalten von Flächen beschränkter mittlerer Krümmung anC 1-Randkurven. Nachr. Akad. Wiss. Göttingen, Math.-Phys. Klasse, Nr. 2 (1979).

  9. [Gr]

    Grüter, M.: Über die Regularität schwacher Lösungen des Systems\((\mathfrak{x})\mathfrak{x}_u \wedge \mathfrak{x}_v \). Dissertation, Universität Düsseldorf (1979).

  10. [Ha]

    Hartman, P., &Wintner, A.: On the local behavior of solutions of nonparabolic partial differential equations. Am. J. of Math.75, 449–476 (1953).

  11. [H 1]

    Heinz, E.: An inequality of isoperimetric type for surfaces of constant mean curvature. Arch. Rational Mech. Anal.33, 155–168 (1969).

  12. [H 2]

    Heinz, E.: Über das Randverhalten quasilinearer elliptischer Systeme mit isothermen Parametern. Math. Z.113, 99–105 (1970).

  13. [M]

    Meeks, W. H., &Yau, S. T.: The existence of embedded minimal surfaces and the problem of uniqueness. Math. Z.179, 151–168 (1982).

  14. [N]

    Nitsche, J. C. C.: Contours bounding at most finitely many solutions of Plateau's Problem. Appeared in:Bogoljubov, N. N. (Ed.): Publication in honour ofI. N. Vekua, Moscow 1978, pp. 438–446.

  15. [Q]

    Quien, N.: Über die endliche Lösbarkeit des Plateau-Problems in Riemannschen Mannigfaltigkeiten. Manuscr. math.39, 313–338 (1982).

  16. [R]

    Ruchert, H.: Ein Eindeutigkeitssatz für Flächen konstanter mittlerer Krümmung. Archiv der Math.33, 91–104 (1979).

  17. [S 1]

    Sauvigny, F.: Flächen vorgeschriebener mittlerer Krümmung mit eineindeutiger Projektion auf eine Ebene. Math. Z.180, 41–67 (1982).

  18. [S 2]

    Sauvigny, F.: A-priori-Abschätzungen der Hauptkrümmungen für Immersionen vom Mittleren-Krümmungs-Typ mittels Uniformisierung und Sätze vom Bernstein-Typ. Habilitationsschrift, Georg-August-Universität Göttingen, 1989.

  19. [S 3]

    Sauvigny, F.: A priori estimates of the principal curvatures for immersions of prescribed mean curvature and theorems of Bernstein type. Submitted to Math. Z.

  20. [sc]

    Schoen, R.: Estimates for stable minimal surfaces in three dimensional manifolds. Appeared inE. Bombieri (Ed.): Seminar on minimal submanifolds. Princeton: Princeton University Press 1983. Annals of Math. Studies. 103, pp. 111–126.

  21. [T 1]

    Tomi, F.: On the local uniqueness of the problem of least area. Arch. Rational Mech. Anal.52, 312–318 (1973).

  22. [T 2]

    Tomi, F.: On the finite solvability of Plateau's Problem, inJ. Palis &M. Do Carmo (Eds.): Geometry and Topology. Berlin,...: Springer 1977. Lecture Notes in Math. 597, pp. 679–695.

Download references

Author information

Additional information

Communicated by J. C. C.Nitsche

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Sauvigny, F. On immersions of constant mean curvature: Compactness results and finiteness theorems for Plateau's problem. Arch. Rational Mech. Anal. 110, 125–140 (1990).

Download citation


  • Neural Network
  • Complex System
  • Nonlinear Dynamics
  • Electromagnetism
  • Integral Condition