Minimal surfaces in the large

  • Tibor Radó
Part of the Ergebnisse der Mathematik und Ihrer Grenƶgebiete book series (volume 2)

Abstract

The problem of Plateau, as considered for instance by H. A. Schwarz in his classical investigations1, calls for à minimal surface 𝕸 which is bounded by a given Jordan curve Γ* and which is free of singularities in its interior. The question naturally arises if this supplementary condition can be complied with if Γ* is very complicated. The very particular cases which have been dealt with in the older literature certainly cannot be considered as representative of what might be expected in the case of a general Jordan curve. At any rate, that supplementary condition has been dropped, at least for the time being, in the modern general investigations, and the existence of a solution free of singularities has been established only for certain special classes of curves.

Keywords

Sine Dition Topo Aire Carus 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Gesammelte Mathematische Abhandlungen Vol. 1.Google Scholar
  2. 2.
    Gesammelte Mathematische Abhandlungen Vol. 1 p. 111.Google Scholar
  3. 3.
    T. Radó: Contributions to the theory of minimal surfaces. Acta Litt. Sci. Szeged Vol. 6 (1932) pp. 1–20.Google Scholar
  4. 1.
    Surfaces of different topological types will be only considered at the end of Chapter VI.Google Scholar
  5. 2.
    This problem has been called the problem of Plateau by Lebesgue: Intégrale, longueur, aire. Ann. Mat. pura appl. Vol. 7 (1902) pp. 231–359.CrossRefGoogle Scholar
  6. 1.
    T. Radó: Contributions to the theory of minimal surfaces. Acta Litt. Sci. Szeged Vol. 6 (1932) p. 10, where the lemma is stated for n = 2.Google Scholar
  7. †.
    The reviewer learned about this theorem from L. Fejér.Google Scholar
  8. 2.
    The theorem is also true for surfaces with negative curvature. This fact played an important role in the work of S. Bernstein on partial differential equations of the elliptic type. See for references L. Lichtenstein: Neuere Entwicklung usw. Enzyklopädie der math. Wiss. Vol. 2 (3) pp. 1277–1334.Google Scholar
  9. 1.
    See T. Radó: The problem of the least area and the problem of Plateau. Math. Z. Vol. 32 (1930) p. 794, where the theorem is stated for n = 1.CrossRefGoogle Scholar
  10. 2.
    See, for instance, Kerékjórtó: Vorlesungen liber Topologie I, p. 175.Google Scholar
  11. 1.
    See T. Radó: The problem of the least area and the problem of Plateau. Math. Z. Vol. 32 (1930) pp. 763–796.MathSciNetMATHCrossRefGoogle Scholar
  12. 1a.
    T. Radó: Contributions to the theory of minimal surfaces. Acta Litt. Sci. Szeged Vol. 6 (1932) pp. 1–20.Google Scholar
  13. 2.
    See, also for references, the beautiful treatment of this theorem and of related subjects by E. Hopf: Elementare Bemerkungen über die Lösungen partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus. S.-B. preuß. Akad. Wiss. 1927 pp. 147–152.Google Scholar
  14. 2a.
    See also A. Haar: Über reguläre Variationsprobleme. Acta Litt. Sci. Szeged Vol. 3 (1927) pp. 224–234.MATHGoogle Scholar
  15. 3.
    T. Radó: Contributions to the theory of minimal surfaces. Acta Litt. Sci. Szoged Vol.6 (1932) pp. 1–20.Google Scholar
  16. 1.
    Intégrale, longueur, aire. Ann. Mat. pura appl. Vol. 7 (1902) pp. 231–359.Google Scholar
  17. 1.
    See, also for references, the beautiful Chapter IV in the little book of G. A. Bliss: Calculus of Variations (No. l of the Carus Mathematical Monographs).Google Scholar
  18. 2.
    Gesammelte Mathematische Abhandlungen Vol. 1 pp. 151–167 and 223–269.Google Scholar
  19. 3.
    The formulas (3.2) are obtained by choosing λ(w) = w in (2.28).Google Scholar
  20. 1.
    Gesammelte Mathematische Abhandlungen Vol. 1 pp, 241–269.Google Scholar
  21. 2.
    w is the stereographic projection of the spherical image of the surface. See Darboux: Théorie générale des surfaces Vol. i pp. 347–348.Google Scholar
  22. 3.
    T. Radó: Contributions to the theory of minimal surfaces. Acta Litt. Sci. Szeged Vol.6 (1932) pp. 1–20.Google Scholar
  23. 1.
    T. Radó: Contributions to the theory of minimal surfaces. Acta Litt. Sci. Szeged Vol. 6 (1932) pp. 1–20.Google Scholar
  24. 2.
    The formulas (3.4) define the so-called minimal surface of Enneper. See Darboux: Théorie générale des surfaces Vol. 1 pp. 372–376. Examples of a less elementary character have been given by H. A. Schwarz: Gesammelte Mathematische Abhandlungen Vol. 1 pp. 151–167 and 223–269.Google Scholar
  25. 1.
    See G. A. Bliss: Calculus of Variations (No. 1 of the Cams Mathematical Monographs).Google Scholar
  26. 2.
    The same remark applies to an example due to N. Wiener. See J. Douglas: Solution of the problem of Plateau. Trans. Amer. Math. Soc. Vol. 33 (1931) p. 269.CrossRefGoogle Scholar
  27. 3.
    S. Bernstein: Sur les équations du Calcul des Variations. Ann. École norm. Vol.29 (1912) pp. 431–485. See in particular pp. 484–485.MATHGoogle Scholar
  28. 4.
    T. Radó: Contributions to the theory of minimal surfaces. Acta Litt. Sci. Szeged Vol. 6 (1932) pp. 1–20.Google Scholar
  29. 5.
    The uniqueness is absolutely essential for the conclusion that problem P 5 is not solvable. For this reason, several examples presented in the literature are incomplete.Google Scholar
  30. 1.
    Or in the even more severe form based on the formulas (32).Google Scholar
  31. 1.
    We mean the following theorem: if the interior of a Jordan curve is mapped, in a one-to-one and conformai way, upon the interior of the unit circle, then the map remains continuous and one-to-one on the boundaries. See for instance Carathéodory: Conformai representation (Cambridge Tracts 28).Google Scholar
  32. 2.
    This program has been carried out in a joint paper by E. F. Beckenbach and T. Radó: Subharmonic functions and minimal surfaces. To appear in Trans. Amer. Math. Soc.Google Scholar
  33. 3.
    See H.A. Schwarz: Gesammelte Mathematische Abhandlungen Vol. I.Google Scholar
  34. 4.
    Cf. VI. 35.Google Scholar
  35. 1.
    Compare, for instance, the following two papers of H. A. Schwarz: Über einige Abbildungsaufgaben. Gesammelte Mathematische Abhandlungen Vol. 2 pp. 65–83 and Bestimmung einer speziellen Minimalfläche. Gesammelte Mathematische Abhandlungen Vol. 1 pp. 6–125. See also Darboux: Theorie generale des surfaces Vol. 1 pp. 4901601.Google Scholar
  36. 1.
    See J. Douglas Solution of the problem of Plateau. Trans. Amer. Math. Soc. Vol.33 (1931) pp. 263–321.MathSciNetCrossRefGoogle Scholar
  37. 1.
    Zur Theorie der Minimalflächen. Math. Z. Vol. 9 (1921) pp. 154–160.Google Scholar
  38. 2.
    We follow the simple proof given by E. F. Beckenbach: The area and boundary of minimal surfaces. Ann. of Math. Vol. 33 (1932) pp. 658–664.MathSciNetCrossRefGoogle Scholar
  39. 3.
    See for instance Pólya-Szegö: Aufgaben und Lehrsätze Vol. 2 p. 14.Google Scholar
  40. 4.
    T. Carleman: Zur Theorie der Minimalflächen. Math. Z. Vol. 9 (1921) p. 160.MathSciNetCrossRefGoogle Scholar
  41. 1.
    See E. F. Beckenbach: The area and boundary of minimal surfaces. Ann. of Math. Vol. 33 (1932) pp. 658–664. — The theorems in III.24 and III.25 also hold for surfaces of negative curvature, given in isothermic representation. See E. F. Beckenbach and T. Radó: Subharmonic functions and surfaces of negative curvature. To appear in Trans. Amer. Math. Soc.MathSciNetCrossRefGoogle Scholar
  42. 2.
    III.26 to III.29 are taken from E. F. Beckenbach and T. Radó: Subharmonic functions and minimal surfaces. To appear in Trans. Amer. Math. Soc.Google Scholar
  43. 3.
    See F. Riesz: Sur les fonctions subharmoniques etc. Acta Math. Vol. 48 (1926) pp. 329–343.MathSciNetMATHCrossRefGoogle Scholar
  44. 1.
    See for instance Pólya-Szegö: Aufgaben und Lehrsätze Vol. 1 p. 138 problem 277.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1933

Authors and Affiliations

  • Tibor Radó

There are no affiliations available

Personalised recommendations