Annali di Matematica Pura ed Applicata (1923 -)

, Volume 196, Issue 4, pp 1513–1524 | Cite as

A necessary condition for the existence of a doubly connected minimal surface

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Abstract

Given two circles contained in parallel planes, it is expectable that there does not exist a doubly connected minimal surface bounded by both circles if these circles are either laterally or vertically far away. In this paper, we give a quantitative estimate of this separation. We also obtain bounds for the height of a Riemann minimal example in terms of a catenoid with the same boundary radii and waist.

Keywords

Minimal surface Riemann minimal example Elliptic integral 

Mathematics Subject Classification

53A10 53C42 
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References

  1. 1.
    Enneper, A.: Die cyklischen Flächen. Z. Math. Phys. 14, 393–421 (1869)Google Scholar
  2. 2.
    Goldschmidt, B.: Determination superficiei minimae rotatione curvae data duo puncta jungentis circa datum axem ortae. Göttingen (1831)Google Scholar
  3. 3.
    Hildebrandt, S.: Maximum principles for minimal surfaces and for surfaces of continuous mean curvature. Math. Z. 128, 253–269 (1972)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Lindelöf, L.: Sur les limites entre lesquelles le caténoide est une surface minima. Math. Ann. 2, 160–166 (1870)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Meeks III, W.H., White, B.: Minimal surfaces bounded by convex curves in parallel planes. Comment. Math. Helv. 66, 263–278 (1991)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Nitsche, J.C.C.: A Supplement to the condition of J. Douglas. Rend. Circ. Matem. Palermo, Serie II, T13, 659–666 (1964)Google Scholar
  7. 7.
    Nitsche, J.C.C.: A necessary criterion for the existence of certain minimal surfaces. Indiana Univ. Math. J. 13(4), 659–666 (1964)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Nitsche, J.C.C.: On new results in the theory of minimal surfaces. Bull. Am. Math. Soc. 71, 195–270 (1965)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Nitsche, J.C.C.: Ein Einschliessungssatz für Minimalflächen. Math. Ann. 165, 71–75 (1966)CrossRefGoogle Scholar
  10. 10.
    Nitsche, J.C.C.: Note on the nonexistence of minimal surfaces. Proc. Am. Math. Soc. 19, 1303–1305 (1968)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Nitsche, J.C.C.: Lectures on Minimal Surfaces. Cambridge University Press, Cambridge (1989)Google Scholar
  12. 12.
    Nitsche, J.C.C., Leavitt, J.: Numerical estimates for minimal surfaces. Math. Ann. 180, 170–174 (1969)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Osserman, R., Schiffer, M.: Doubly-connected minimal surfaces. Arch. Rational Mech. Anal. 58, 285–307 (1975)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Plateau, J.: Recherches expérimentales et théoriques sur les figures d’équilibre d’une masse liquide sans pesanteur. Mém. Acad. Royale Sci. Belg.; Ser. IV: vol. 31 (1859), Ser. X: vol. 37 (1869)Google Scholar
  15. 15.
    Riemann, B.: Oeuvres mathématiques. Gauthiers-Villars, Paris (1898)MATHGoogle Scholar
  16. 16.
    Rossman, W.: Minimal surfaces with planar boundary curves. Kyushu J. Math. 52, 209–225 (1998)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Shiffman, M.: On surfaces of stationary area bounded by two circles, or convex curves, in parallel planes. Annals Math. 63, 77–90 (1956)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Departamento de Geometría y Topología, Instituto de Matemáticas (IEMath-GR)Universidad de GranadaGranadaSpain

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