Acta Mathematica

, Volume 193, Issue 2, pp 141–174

A Hopf differential for constant mean curvature surfaces inS2×R andH2×R

  • Uwe Abresch
  • Harold Rosenberg


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Abresch, U., Constant mean curvature tori in terms of elliptic functions.J. Reine Angew. Math., 374 (1987), 169–192.MATHMathSciNetGoogle Scholar
  2. [2]
    —, Old and new doubly periodic solutions of the sinh-Gordon equation, inSeminar on New Results in Nonlinear Partial Differential Equations (Bonn, 1984), pp. 37–73. Aspects Math., E10. Vieweg, Braunschweig, 1987.Google Scholar
  3. [3]
    Alexandrov, A. D., Uniqueness theorems for surfaces in the large, V.Vestnik Leningrad. Univ., 13:19 (1958), 5–8 (Russian).MathSciNetGoogle Scholar
  4. [4]
    —, Uniqueness theorems for surfaces in the large, I–V.Amer. Math. Soc. Transl., 21 (1962), 341–416.MathSciNetGoogle Scholar
  5. [5]
    —, A characteristic property of spheres.Ann. Mat. Pura Appl., 58 (1962), 303–315.MathSciNetGoogle Scholar
  6. [6]
    Bobenko, A. I., All constant mean curvature tori inR 3,S 3,H 3 in terms of theta-functions.Math. Ann., 290 (1991), 209–245.CrossRefMATHMathSciNetGoogle Scholar
  7. [7]
    Burstall, F. E., Ferus, D., Pedit, F. &Pinkall, U., Harmonic tori in symmetric spaces and commuting Hamiltonian systems on loop algebras.Ann. of Math., 138 (1993), 173–212.CrossRefMathSciNetGoogle Scholar
  8. [8]
    Collin, P., Hauswirth, L. &Rosenberg, H., The geometry of finite topology Bryant surfaces.Ann. of Math., 153 (2001), 623–659.MathSciNetGoogle Scholar
  9. [9]
    Figueroa, C. B., Mercuri, F. &Pedrosa, R. H. L., Invariant surfaces of the Heisenberg groups.Ann. Mat. Pura Appl., 177 (1999), 173–194.MathSciNetGoogle Scholar
  10. [10]
    Gauss, C. F., Allgemeine Auflösung der Aufgabe: Die Theile einer gegebenen Fläche auf einer andern gegebenen Fläche so abzubilden, dass die Abbildung dem Abgebildeten in der Kleinsten Theilen, ähnlich wird (Kopenhagener Preisschrift).Astron. Abhandl., 3 (1825), 1–30;Phil. Mag., 4 (1828), 104–113, 206–215;Carl Friedrich Gauss Werke, Vierter Band, pp. 189–216. Der Königlichen Gesellschaft der Wissenschaften, Göttingen, 1873.Google Scholar
  11. [11]
    Hauswirth, L., Roitman, P. &Rosenberg, H., The geometry of finite topology Bryant surfaces quasi-embedded in a hyperbolic manifold.J. Differential Geom., 60 (2002), 55–101.MathSciNetGoogle Scholar
  12. [12]
    Hitchin, N. J., Harmonic maps from a 2-torus to the 3-sphere.J. Differential Geom., 31 (1990), 627–710.MATHMathSciNetGoogle Scholar
  13. [13]
    Hopf, H.,Differential Geometry in the Large. Lecture Notes in Math., 1000. Springer, Berlin, 1983.Google Scholar
  14. [14]
    Hsiang, W.-T. &Hsiang, W.-Y., On the uniqueness of isoperimetric solutions and imbedded soap bubbles in noncompact symmetric spaces, I.Invent. Math., 98 (1989), 39–58.CrossRefMathSciNetGoogle Scholar
  15. [15]
    Hsiang, W.-Y., On soap bubbles and isoperimetric regions in noncompact symmetric spaces, I.Tôhoku Math. J., 44 (1992), 151–175.MATHMathSciNetGoogle Scholar
  16. [16]
    Kapouleas, N., Complete embedded minimal surfaces of finite total curvature.J. Differential Geom., 47 (1997), 95–169.MATHMathSciNetGoogle Scholar
  17. [17]
    de Lira, J., To appear.Google Scholar
  18. [18]
    Mazzeo, R. &Pacard, F., Constant mean curvature surfaces with Delaunay ends.Comm. Anal. Geom., 9 (2001), 169–237.MathSciNetGoogle Scholar
  19. [19]
    Nelli, B. &Rosenberg, H., Minimal surfaces inH 2 ×R.Bull. Braz. Math. Soc., 33 (2002), 263–292.CrossRefMathSciNetGoogle Scholar
  20. [20]
    Pedrosa, R. H. L. &Ritoré, M., Isoperimetric domains in the Riemannian product of a circle with a simply connected space form and applications to free boundary problems.Indiana Univ. Math. J., 48 (1999), 1357–1394.CrossRefMathSciNetGoogle Scholar
  21. [21]
    Pinkall, U. &Sterling, I., On the classification of constant mean curvature tori.Ann. of Math., 130 (1989), 407–451.CrossRefMathSciNetGoogle Scholar
  22. [22]
    Rosenberg, H., Bryant surfaces, inThe Global Theory of Minimal Surfaces in Flat Spaces (Martina Franca, 1999), pp. 67–111. Lecture Notes in Math., 1775. Springer, Berlin, 2002.Google Scholar
  23. [23]
    —, Minimal surfaces inM 2 ×R.Illinois J. Math., 46 (2002), 1177–1195.MATHMathSciNetGoogle Scholar
  24. [24]
    Wells, R. O.,Differential Analysis on Complex Manifolds, 2nd edition. Graduate Texts in Math., 65. Springer, New York-Berlin, 1980.Google Scholar
  25. [25]
    Wente, H., Counterexample to a conjecture of H. Hopf.Pacific J. Math., 121 (1986), 193–243.MATHMathSciNetGoogle Scholar

Copyright information

© Institut Mittag-Leffler 2004

Authors and Affiliations

  • Uwe Abresch
    • 1
  • Harold Rosenberg
    • 2
  1. 1.Fakultät für MathematikRuhr-Universität BochumBochumGermany
  2. 2.Centre de Mathématiques de JussieuUniversité de Paris VII-Denis DiderotParis Cedex 05France

Personalised recommendations