On the Size of a Stable Minimal Surface in R3

  • J. L. Barbosa
  • M. do Carmo
Chapter

Abstract

Let M be a two’dimensional, orientable C∞-manifold. A domain \({D} \subset {M} \) is an open, connected subset with compact closure \({D} \subset {M} \) and such that the boundary \({\partial}{D}\) is a finite union of piece-wise smooth curves.

Keywords

Branch Point Minimal Immersion Holomorphic Curve Local Diffeomorphism Spherical Image 
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References

  1. [1]
    M. Berger, P. Gauduchon, and E. Mazet, "Le spectre d’une variété Riemannienne," Lecture Notes in Math. No. 194, Springer-Verlag, Berlin, 1971.Google Scholar
  2. [2]
    R. C. Gunning, "Lectures on Riemann Surfaces," Princeton Math. Notes, Princeton Univ. Press, Princeton. New Jersey, 1966.Google Scholar
  3. [3]
    W. K. Hayman, and S. Friedland, "Eigenvalue inequalities for the Dirichlet problem on the sphere and the growth of subharmonic functions," to appear.Google Scholar
  4. [4]
    L. Uchtenstein, "Beiträge zur theorie der linearen partiellen differentialgleichungen zwelter ordnung von elliptischen typus. Rend. Circ. Mat. Palermo 23 (1912), 201-211.Google Scholar
  5. [5]
    L. Lindelöf, "Sur les llmites entre lesquelles le caténoide est une surface minima," Math. Ann., 2 (1870), 160-166.Google Scholar
  6. [6]
    J. Peetre, "A generalization of Courant’s nodal domain theorem," Math. Scand., S (1957), 15-20.Google Scholar
  7. [7]
    T. Radó, On the Problem of Plateau, Springer-Verlag, New York, Heidelberg. Berlin 1971.Google Scholar
  8. [8]
    A. H. Schwarz, Gesammelte Math. Abhandlungen, Erster Band, J. Springer, Berlin, 1890, 224-269 and 151-167.Google Scholar
  9. [9]
    J. Simons, "Minimal varieties in Riemannian manifolds," Annuals of Math., 88, (1968), 62-105.Google Scholar
  10. [10]
    S. Smale, "On the Morse index theorem," Journal of Math. and Mech., 14 (1965), 1049-1056.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • J. L. Barbosa
    • 1
  • M. do Carmo
    • 1
  1. 1.Instituto de Matematica Pura e AplicadaRio de JaneiroBrasil

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