Acta Mathematica

The existence of minimal surfaces of given topological structure under prescribed boundary conditions

R. Courant

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Abstract

The purpose of the present paper is the solution of the boundary value problems for minimal surfaces when the boundaries are not, or not entirely-fixed Jordan curves but are free to move on prescribed manifolds. At the same time I shall present modifications and simplifications of my previous solution of the Plateau' and Douglas' problem for fixed boundary curves and prescribed topological structure and incidentally discuss certain features of the problem in order to clarify its relation to the theory of conformal mapping. Though based on previous publications, the paper may, except for some references, be read independently.

Note

Nos. [9], [10], [11], [12], [13], [14] of the bibliography at the end of the paper. References to this list are made in square brackets throughout this paper.

Article information

Source
Acta Math., Volume 72 (1940), 51-98.

Dates
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485888312

Digital Object Identifier
doi:10.1007/BF02546328

Mathematical Reviews number (MathSciNet)
MR2478

Zentralblatt MATH identifier
66.0485.02

Rights
1940 © Almqvist & Wiksells Boktryckeri-A.-B.

Citation

Courant, R. The existence of minimal surfaces of given topological structure under prescribed boundary conditions. Acta Math. 72 (1940), 51--98. doi:10.1007/BF02546328. https://projecteuclid.org/euclid.acta/1485888312


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Bibliographical References

  • Radó, T., The problem of Plateau, Ergebnisse der Math. Berlin, 1933, an illuminating report, discussing the preceding literature.
  • Douglas, J., The Problem of Plateau, Bull Am. Math. Soc. 1933, p. 227 ff. containing a complete reference to Douglas' previous papers.
  • —, Some New results in the problem of Plateau, Journal of Math. and Phys., vol. 15 (1936), p. 55 ff.
  • —, Minimal surfaces of general topological structure, Ibid., pp. 106 ff.
  • —, Minimal surfaces of higher topological structure, Proc. Nat. Ac. Sciences, Washington, vol. 24 (1938), pp. 343 ff.
  • —, Greens function and the problem of Plateau, Ibid., p. 353 ff.
  • —, The most general form of the problem of Plateau, Ibid. p. 360 ff.
  • —, Minimal surfaces of higher topological structure, Ann. of Math., vol. 40 (1939), pp. 205–298. Contains detailed bibliography of Douglas' previous papers.
  • Courant, On the Problem of Plateau, Proc. Nat. Ac. Sciences, Wash., vol. 22 (1936), pp. 367 ff.
  • —, Plateaus Problem and Dirichlet's Principle, Ann. of Math., vol. 38 (1937), pp. 679 ff.
  • Courant, and Hilbert, Methoden der Mathematischen Physik., vol. 2 (1937), Cap. 7.
  • Courant, Conformal mapping of multiply connected domains, Duke Math. Journal, vol. 5 (1939).
  • —, The Existence of a Minimal Surface of least area bounded by prescribed Jordan arcs and prescribed surfaces, Proc. Nat. Ac. Sciences, Wash., vol. 24 (1938), p. 97 ff.
  • —, Remarks on Plateau's and Douglas' Problem, Proc. Nat. Acad. Sciences, Wash. vol. 24 (1938), p. 519 ff.
  • Courant, The Dirichlet Principle, Cambridge Tract in preparation.
  • Shiffman, Minimal surfaces of higher topological structure. To appear in Americ. Journal of Math. (1939).
  • —, The Plateau Problem for Minimal Surfaces which are relative Minima. Ann. of Math., vol. 39 (1938), pp. 309 ff.
  • —, Abstract in Bulletin of the Americ. Math. Soc., vol. 44 (1938), p. 637, of a paper read before the society on Sept. 7, 1938.
  • Shiffman, The Plateau Problem for Non-relative Minima. Proc. Nat. Acad. Sciences, vol. 25 (1939), p. 215–220.
  • Shiffman, The Plateau Problem for Non-relative Minima. To appear in Ann. of Math.
  • Morse and Tompkins, Existence of Minimal Surfaces of General Critical Type. Proc. Nat. Acad. Wash., vol. 25 (1939), p. 153 ff.
  • ——, Same Title. Ann. of Math., vol. 49 (1939), p. 443 ff.
  • I. Ritter, Solution of Schwarz' Problem concerning Minimal Surfaces. Not yet published.