Commentarii Mathematici Helvetici

, Volume 65, Issue 1, pp 255–270 | Cite as

The maximum principle at infinity for minimal surfaces in flat three manifolds

  • William H. MeeksIII
  • Harold Rosenberg
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References

  1. [1]
    T. Choi, W. H. Meeks III andB. White,A rigidity theorem for, properly embedded minimal surfaces in3. Journal of Differential Geometry, March, 1990.Google Scholar
  2. [2]
    M. do Carmo andC. K. Peng, Stable minimal surfaces in ℝ3 are planes. Bulletin of the AMS1, 903–906 (1979).MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    D. Fischer-Colbrie,On complete minimal surfaces with finite Morse index in 3-manifolds. Inventiones Math.82, 121–132 (1985).MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    D. Fischer-Colbrie andR. Schoen,The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature. Comm. Pure Appl. Math.33, 199–211 (1980).MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    C. Frohman andW. H. Meeks III,The topological uniqueness of complete one-ended minimal surfaces and Heegaard surfaces in3 (preprint).Google Scholar
  6. [6]
    R. Hardt andL. Simon,Boundary regularity and embedded minimal solutions for the oriented Plateau problem. Annals of Math.110, 439–486 (1979).MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    D. Hoffman andW. H. Meeks III,The strong halfspace theorem for minimal surfaces. Inventiones Math. (to appear).Google Scholar
  8. [8]
    D. Hoffman andW. H. Meeks III,The asymptotic behavior of properly embedded minimal surfaces of finite topology. Journal of the AMS2, 667–681 (1989).MathSciNetMATHGoogle Scholar
  9. [9]
    R. Langevin andH. Rosenberg,A maximum principle at infinity for minimal surfaces and applications. Duke Math. Journal57, 819–828, (1988).MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    S. Lojasiewicz,Triangulation of semianalytic sets. Ann. Scuola Norm. Sup. Pisa18, 449–474 (1964).MathSciNetMATHGoogle Scholar
  11. [11]
    W. H. Meeks III,The topological uniqueness of minimal surfaces in three-dimensional Euclidean space. Topology20, 389–410 (1981).MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    W. H. Meeks III andH. Rosenberg,The geometry of periodic minimal surfaces (preprint).Google Scholar
  13. [13]
    W. H. Meeks III andH. Rosenberg,The global theory of doubly periodic minimal surfaces. Inventiones Math.97, 351–379 (1989).MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    W. H. Meeks III, L. Simon andS. T. Yau,The existence of embedded minimal surfaces, exotic spheres and positive Ricci curvature. Annals of Math.116, 221–259 (1982).MathSciNetGoogle Scholar
  15. [15]
    W. H. Meeks III andS. T. Yau,The topological uniqueness theorem of complete minimal surfaces of finite topological type (preprint).Google Scholar
  16. [16]
    W. H. Meeks III andS. T. Yau,The existence of embedded minimal surfaces and the problem of uniqueness. Math. Z.179, 151–168 (1982).MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    R. Osserman,A Survey of Minimal Surfaces, 2nd edition. Dover Publications, New York, 1986.MATHGoogle Scholar
  18. [18]
    M. Protter andH. Weinberger,Maximum Principles in Differential Equations, Prentice-Hall, Englewood 1967.MATHGoogle Scholar
  19. [19]
    R. Schoen,Estimates for Stable Minimal Surfaces in Three Dimensional Manifolds, volume 103 ofAnnals of Math. Studies. Princeton University Press, 1983.Google Scholar
  20. [20]
    R. Schoen,Uniqueness, symmetry, and embeddedness of minimal surfaces. Journal of Differential Geometry18, 791–809 (1983).MathSciNetMATHGoogle Scholar
  21. [21]
    L. Simon,Lectures on geometric measure theory. Inproceedings of the Center for Mathematical Analysis, volume 3. Australian National University, Canberra 1983.Google Scholar
  22. [22]
    J. A. Wolf,Spaces of Constant Curvature. McGraw-Hill, New York 1967.MATHGoogle Scholar

Copyright information

© Birkhäuser Verlag 1990

Authors and Affiliations

  • William H. MeeksIII
    • 1
    • 2
  • Harold Rosenberg
    • 1
    • 2
  1. 1.Mathematics DepartmentUniversity of MassachusettsAmherstUSA
  2. 2.Department de MathématiqueUniversité de Paris 7ParisFrance

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