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Surfaces with Constant Mean Curvature

  • Rafael López
Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

In this chapter we shall review some basic aspects of the theory of surfaces with constant mean curvature. Surfaces with constant mean curvature will arise as solutions of a variational problem associated to the area functional and related with the classical isoperimetric problem. We state the first and second variation formula for the area and we give the notion of stability of a cmc surface. Next, we introduce the complex analysis as a basic tool in the theory and this will allow to prove the Hopf theorem. Also, we compute the Laplacians of some functions that contain geometric information of a cmc surface. As the Laplacian operator is elliptic, the maximum principle yields height estimates of a graph of constant mean curvature. Finally, and with the aid of the expression of these Laplacians, we will derive the Barbosa-do Carmo theorem that characterizes a round sphere in the family of closed stable cmc surfaces of Euclidean space.

Keywords

Minimal Surface Compact Surface Jacobi Operator Round Sphere Isoperimetric Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. [Ale56]
    Alexandrov, A.D.: Uniqueness theorems for surfaces in the large I–V. Vestn. Leningr. Univ. 11 #19, 5–17 (1956); 12 #7, 15–44, 1957; 13 #7, 14–26, 1958; 13 #13, 27–34, 1958; 13 #19, 5–8, (1958). English transl. in Amer. Math. Soc. Transl. 21, 341–354, 354–388, 389–403, 403–411, 412–416 (1962) Google Scholar
  2. [Ale62]
    Alexandrov, A.D.: A characteristic property of spheres. Ann. Mat. Pura Appl. 58, 303–315 (1962) MathSciNetCrossRefGoogle Scholar
  3. [BC84]
    Barbosa, J.L., do Carmo, M.: Stability of hypersurfaces with constant mean curvature. Math. Z. 185, 339–353 (1984) MathSciNetzbMATHCrossRefGoogle Scholar
  4. [Car92]
    do Carmo, M.: Riemannian Geometry. Birkhäuser, Boston (1992) zbMATHCrossRefGoogle Scholar
  5. [Chv94]
    Chavel, I.: Riemannian Geometry: A Modern Introduction. Cambridge University Press, New York (1994) Google Scholar
  6. [CH89]
    Courant, R., Hilbert, D.: Methods of Mathematical Physics. Wiley, New York (1989) CrossRefGoogle Scholar
  7. [Fed69]
    Federer, H.: Geometric Measure Theory. Springer, New York (1969) zbMATHGoogle Scholar
  8. [FS80]
    Fischer-Colbrie, D., Schoen, R.: The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvature. Commun. Pure Appl. Math. 33, 199–211 (1980) MathSciNetzbMATHCrossRefGoogle Scholar
  9. [GM00a]
    Gálvez, J.A., Martínez, A.: The Gauss map and second fundamental form of surfaces in R 3. Geom. Dedic. 81, 181–192 (2000) zbMATHCrossRefGoogle Scholar
  10. [GM00b]
    Gálvez, J.A., Martínez, A.: Estimates in surfaces with positive constant Gauss curvature. Proc. Am. Math. Soc. 128, 3655–3660 (2000) zbMATHCrossRefGoogle Scholar
  11. [GT01]
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001). Reprint of the 1998 edition zbMATHGoogle Scholar
  12. [Gro96]
    Grosse-Brauckmann, K.: Stable constant mean curvature surfaces minimize area. Pac. J. Math. 175, 527–534 (1996) MathSciNetzbMATHGoogle Scholar
  13. [Hop83]
    Hopf, H.: Differential Geometry in the Large. Lecture Notes in Mathematics, vol. 1000. Springer, Berlin (1983) zbMATHCrossRefGoogle Scholar
  14. [Jel53]
    Jellet, J.H.: Sur la surface dont la courbure moyenne est constant. J. Math. Pures Appl. 18, 163–167 (1853) Google Scholar
  15. [Koi02]
    Koiso, M.: Deformation and stability of surfaces with constant mean curvature. Tohoku Math. J. 54, 145–159 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  16. [KKMS92]
    Korevaar, N., Kusner, R., Meeks, W. III, Solomon, B.: Constant mean curvature surfaces in hyperbolic space. Am. J. Math. 114, 1–43 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  17. [Lop03b]
    López, R.: Some a priori bounds for solutions of the constant Gauss curvature equation. J. Differ. Equ. 194, 185–197 (2003) zbMATHCrossRefGoogle Scholar
  18. [Lop07]
    López, R.: On uniqueness of graphs with constant mean curvature. J. Math. Kyoto Univ. 46, 771–787 (2007) Google Scholar
  19. [Lop09]
    López, R.: A new proof of a characterization of small spherical caps. Results Math. 55, 427–436 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  20. [Mee88]
    Meeks, W.H. III: The topology and geometry of embedded surfaces of constant mean curvature. J. Differ. Geom. 27, 539–552 (1988) MathSciNetzbMATHGoogle Scholar
  21. [Mor95]
    Morgan, F.: Geometric Measure Theory: A Beginner’s Guide, 2nd edn. Academic Press, San Diego (1995) zbMATHGoogle Scholar
  22. [Mul02]
    Müller, F.: Analyticity of solutions for semilinear elliptic systems of second order. Calc. Var. Partial Differ. Equ. 15, 257–288 (2002) zbMATHCrossRefGoogle Scholar
  23. [Rei82]
    Reilly, R.: Mean curvature, the Laplacian, and soap bubbles. Am. Math. Mon. 89, 180–188 (1982) and 197–198 MathSciNetzbMATHCrossRefGoogle Scholar
  24. [Rsb93]
    Rosenberg, H.: Hypersurfaces of constant curvature in space forms. Bull. Sci. Math. 117, 211–239 (1993) MathSciNetzbMATHGoogle Scholar
  25. [Ser69a]
    Serrin, J.: On surfaces of constant mean curvature which span a given space curve. Math. Z. 112, 77–88 (1969) MathSciNetzbMATHCrossRefGoogle Scholar
  26. [Tom75]
    Tomi, F.: A perturbation theorem for surfaces of constant mean curvature. Math. Z. 141, 253–264 (1975) MathSciNetzbMATHCrossRefGoogle Scholar
  27. [Wen91]
    Wente, H.C.: A note on the stability theorem of J.L. Barbosa and M. do Carmo for closed surfaces of constant mean curvature. Pac. J. Math. 147, 375–379 (1991) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Rafael López
    • 1
  1. 1.Department of Geometry and TopologyUniversity of GranadaGranadaSpain

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