Acta Mathematica

A Hopf differential for constant mean curvature surfaces in S2×R and H2×R

Uwe Abresch and Harold Rosenberg

Full-text: Open access

Dedication

Dedicated to Hermann Karcher on the occasion of his 65th birthday

Article information

Source
Acta Math., Volume 193, Number 2 (2004), 141-174.

Dates
Received: 21 January 2004
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485891698

Digital Object Identifier
doi:10.1007/BF02392562

Mathematical Reviews number (MathSciNet)
MR2134864

Rights
2004 © Institut Mittag-Leffler

Citation

Abresch, Uwe; Rosenberg, Harold. A Hopf differential for constant mean curvature surfaces in S 2 ×R and H 2 ×R. Acta Math. 193 (2004), no. 2, 141--174. doi:10.1007/BF02392562. https://projecteuclid.org/euclid.acta/1485891698


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References

  • Abresch, U., Constant mean curvature tori in terms of elliptic functions. J. Reine Angew. Math., 374 (1987), 169–192.
  • —, Old and new doubly periodic solutions of the sinh-Gordon equation, in Seminar on New Results in Nonlinear Partial Differential Equations (Bonn, 1984), pp. 37–73. Aspects Math., E10. Vieweg, Braunschweig, 1987.
  • Alexandrov, A. D., Uniqueness theorems for surfaces in the large, V. Vestnik Leningrad. Univ., 13:19 (1958), 5–8 (Russian).
  • —, Uniqueness theorems for surfaces in the large, I–V. Amer. Math. Soc. Transl., 21 (1962), 341–416.
  • —, A characteristic property of spheres. Ann. Mat. Pura Appl., 58 (1962), 303–315.
  • Bobenko, A. I., All constant mean curvature tori in R3, S3, H3 in terms of theta-functions. Math. Ann., 290 (1991), 209–245.
  • 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.
  • Collin, P., Hauswirth, L. & Rosenberg, H., The geometry of finite topology Bryant surfaces. Ann. of Math., 153 (2001), 623–659.
  • Figueroa, C. B., Mercuri, F. & Pedrosa, R. H. L., Invariant surfaces of the Heisenberg groups. Ann. Mat. Pura Appl., 177 (1999), 173–194.
  • 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.
  • 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.
  • Hitchin, N. J., Harmonic maps from a 2-torus to the 3-sphere. J. Differential Geom., 31 (1990), 627–710.
  • Hopf, H., Differential Geometry in the Large. Lecture Notes in Math., 1000. Springer, Berlin, 1983.
  • 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.
  • Hsiang, W.-Y., On soap bubbles and isoperimetric regions in noncompact symmetric spaces, I. Tôhoku Math. J., 44 (1992), 151–175.
  • Kapouleas, N., Complete embedded minimal surfaces of finite total curvature. J. Differential Geom., 47 (1997), 95–169.
  • de Lira, J., To appear.
  • Mazzeo, R. & Pacard, F., Constant mean curvature surfaces with Delaunay ends. Comm. Anal. Geom., 9 (2001), 169–237.
  • Nelli, B. & Rosenberg, H., Minimal surfaces in H2×R. Bull. Braz. Math. Soc., 33 (2002), 263–292.
  • 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.
  • Pinkall, U. & Sterling, I., On the classification of constant mean curvature tori. Ann. of Math., 130 (1989), 407–451.
  • Rosenberg, H., Bryant surfaces, in The Global Theory of Minimal Surfaces in Flat Spaces (Martina Franca, 1999), pp. 67–111. Lecture Notes in Math., 1775. Springer, Berlin, 2002.
  • —, Minimal surfaces in M2×R. Illinois J. Math., 46 (2002), 1177–1195.
  • Wells, R. O., Differential Analysis on Complex Manifolds, 2nd edition. Graduate Texts in Math., 65. Springer, New York-Berlin, 1980.
  • Wente, H., Counterexample to a conjecture of H. Hopf. Pacific J. Math., 121 (1986), 193–243.