Skip to main content
Log in

Maximum principles for minimal surfaces and for surfaces of continuous mean curvature

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

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

Access this article

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

Instant access to the full article PDF.

Bibliography

  1. Beckenbach, E., Radó, T.: Subharmonic functions and minimal surfaces. Trans. Amer. Math. Soc.35 648–661 (1933).

    Google Scholar 

  2. Bliss, G. A.: Calculus of Variations. Open Court Publ. Co. La Salle 1925.

    Google Scholar 

  3. Courant, R.: Dirichlet's principle, conformal mapping, and minimal surfaces. New York: Interscience Publ. 1950.

    Google Scholar 

  4. Delauney, C.: Sur la surface de revolution dont la courbure moyenne est constante. J. Math. Pures Appl.6, 309–315 (1841).

    Google Scholar 

  5. Dombrowski, P.: Krümmungsgrößen gleichungsdefinierter Untermannigfaltigkeiten Riemannscher Mannigfaltigkeiten. Math. Nachr.38, 133–180 (1968).

    Google Scholar 

  6. Douglas, J.: The problem of Plateau for two contours. J. Math. Phys.10, 315–359 (1931).

    Google Scholar 

  7. Douglas, J.: Minimal surfaces of higher topological structure. Ann. of Math.40, 205–298 (1939).

    Google Scholar 

  8. Gulliver, R. D.: The Plateau problem for surfaces of prescribed mean curvature in a Riemannian manifold. To appear.

  9. Gulliver, R. D., Osserman, R., Royden, H. L.: A theory of branched immersions of surfaces. Preprint.

  10. Gulliver, R. D., Spruck, J.: Surfaces of constant mean curvature which have a simple projection. Math. Z., to appear.

  11. Heinz, E.: Über die Existenz einer Fläche konstanter mittlerer Krümmung bei vorgegebener Berandung. Math. Ann.127, 258–287 (1954).

    Google Scholar 

  12. Heinz, E.: On the nonexistence of a surface of constant mean curvature with finite area and prescribed rectifiable boundary. Arch. Rat. Mech. Analysis35, 249–252 (1969).

    Google Scholar 

  13. Heinz, E., Hildebrandt, S.: The number of branch points of surfaces of bounded mean curvature. J. Diff. Geometry4, 227–235 (1970).

    Google Scholar 

  14. Heinz, E., Hildebrandt, S.: Some remarks on minimal surfaces in Riemannian manifolds. Commun. Pure Appl. Math.23, 371–377 (1970).

    Google Scholar 

  15. Hildebrandt, S.: Einige Bemerkungen über Flächen beschränkter mittlerer Krümmung. Math. Z.115, 169–178 (1970).

    Google Scholar 

  16. Hildebrandt, S.: On the regularity of solutions of two-dimensional variational problems with obstructions. Commun. Pure Appl. Math., to appear.

  17. Hildebrandt, S., Kaul, H.: Two-dimensional variational problems with obstructions and Plateau's problem forH-surfaces in a Riemannian manifold. Commun. Pure Appl. Math.25, 187–223 (1972).

    Google Scholar 

  18. Kaul, H.: Isoperimetrische Ungleichung und Gauß-Bonnet-Formel fürH-Flächen in Riemannschen Mannigfaltigkeiten. Arch. Rat. Mech. Analysis45, 194–221 (1972).

    Google Scholar 

  19. Kaul, H.: Ein Einschließungssatz fürH-Flächen in Riemannschen Mannigfaltigkeiten. Manuscripta Math.5, 103–112 (1971).

    Google Scholar 

  20. Lawson, B.: The global behavior of minimal surfaces inS″. Ann. of Math.92, 224–237 (1970).

    Google Scholar 

  21. Nitsche, J. C. C.: On new results in the theory of minimal surfaces. Bull. Amer. Math. Soc.71, 195–270 (1965).

    Google Scholar 

  22. Nitsche, J. C. C.: A necessary criterion for the existence of certain minimal surfaces. J. Math. Mech.13, 659–665 (1964).

    Google Scholar 

  23. Nitsche, J. C. C.: A supplement to the condition of J. Douglas Rend. Circ. Mat. Palermo (2)13, 192–198 (1964).

    Google Scholar 

  24. Nitsche, J. C. C.: Ein Einschließungssatz für Minimalflächen. Math. Ann.165, 71–75 (1966).

    Google Scholar 

  25. Nitsche, J. C. C.: Note on the non-existence of minimal surfaces. Proc. Amer. Math. Soc.19, 1303–1305 (1968).

    Google Scholar 

  26. Nitsche, J. C. C., Leavitt, J.: Numerical estimates for minimal surfaces. Math. Ann.180, 170–174 (1969).

    Google Scholar 

  27. Serrin, J.: On surfaces of constant mean curvature which span a given space curve. Math. Z.112, 77–88 (1969).

    Google Scholar 

  28. Sinclair, E.: On the minimum surface of revolution in the case of one variable end point. Ann. of Math.8, 177–188 (1906-1907).

    Google Scholar 

  29. Tomi, F.: Variationsprobleme vom Dirichlet-Typ mit einer Ungleichung als Nebenbedingung. Math. Z., to appear.

  30. Thompson, D'Arcy W.: On growth and form. Cambridge University Press 1969.

  31. Werner, H.: Das Problem von Douglas für Flächen konstanter mittlerer Krümmung. Math. Ann.133, 303–319 (1957).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hildebrandt, S. Maximum principles for minimal surfaces and for surfaces of continuous mean curvature. Math Z 128, 253–269 (1972). https://doi.org/10.1007/BF01111709

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01111709

Keywords

Navigation