Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Constant mean curvature surfaces in $ M^2\times \mathbf{R}$

Authors: David Hoffman, Jorge H. S. de Lira and Harold Rosenberg
Journal: Trans. Amer. Math. Soc. 358 (2006), 491-507
MSC (2000): Primary 53C27, 58J60
Published electronically: September 26, 2005
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The subject of this paper is properly embedded $ H-$surfaces in Riemannian three manifolds of the form $ M^2\times \mathbf{R}$, where $ M^2$ is a complete Riemannian surface. When $ M^2={\mathbf R}^2$, we are in the classical domain of $ H-$surfaces in $ {\mathbf R}^3$. In general, we will make some assumptions about $ M^2$ in order to prove stronger results, or to show the effects of curvature bounds in $ M^2$ on the behavior of $ H-$surfaces in $ M^2\times \mathbf{R}$.

References [Enhancements On Off] (What's this?)

  • 1. J. L. Barbosa and R.S. Earp.
    Prescribed mean curvature in $ H\sp{n+1}(-1)$ with convex planar boundary, I.
    Geom. Dedicata, 71(1):61-74, 1998. MR 1624726 (99d:53064)
  • 2. M. Crandall and P. Rabinowitz.
    Bifurcation, perturbation of simple eigenvalues and linearized stability.
    Arch. Rational Mech. Anal., 52:161--180, 1973. MR 0341212 (49:5962)
  • 3. P. Collin.
    Topologie et courboure des surfaces minimales proprement plongées de $ {\mathbf {R} ^3}$.
    Ann. of Mathematics, 145:1-31, 1997. MR 1432035 (98d:53010)
  • 4. D. Gilbarg and N. S. Trudinger.
    Elliptic partial differential equations of second order.
    Springer-Verlag, New York, 2nd edition, 1983. MR 0737190 (86c:35035)
  • 5. Karsten Grosse-Braukmann and Meinhard Wohlgemuth.
    The gyroid is embedded and has constant mean curvature companions.
    Calc. Var., 4:499-523, 1996. MR 1415998 (97k:53011)
  • 6. D. Hoffman and H. Karcher.
    Complete embedded minimal surfaces of finite total curvature.
    In Encyclopedia of Mathematics, pages 5-93, 1997.
    R. Osserman, editor, Springer Verlag. MR 1490038 (98m:53012)
  • 7. D. Hoffman and W. H. Meeks III.
    Embedded minimal surfaces of finite topology.
    Annals of Mathematics, 131:1-34, 1990. MR 1038356 (91i:53010)
  • 8. W.-T. Hsiang and W.-Y. Hsiang.
    On the uniqueness of isoperimetric solutions and imbedded soap bubbles in non-compact symmetric spaces.
    Inventiones Math. 98:39-58, 1989. MR 1010154 (90h:53078)
  • 9. H. Karcher.
    Embedded minimal surfaces derived from Scherk's examples.
    Manuscripta Math., 62:83-114, 1988. MR 0958255 (89i:53009)
  • 10. H. Karcher.
    The triply periodic minimal surfaces of Alan Schoen and their constant mean curvature companions.
    Manuscripta Math., 64:291-357, 1989. MR 1003093 (90g:53010)
  • 11. N. Korevaar, R. Kusner, and B. Solomon.
    The structure of complete embedded surfaces with constant mean curvature.
    Journal of Differential Geometry, 30:465-503, 1989. MR 1010168 (90g:53011)
  • 12. H. B. Lawson.
    Complete minimal surfaces in $ S^3$.
    Annals of Mathematics, 92:335-374, 1970. MR 0270280 (42:5170)
  • 13. H. B. Lawson, Jr.
    Lectures on Minimal Submanifolds.
    Publish or Perish Press, Berkeley, 1971. MR 0576752 (82d:53035b)
  • 14. R. Mazzeo and F. Pacard.
    Bifurcating nodoids.
    Contemp. Math., 314:169-186, 2002. MR 1941630 (2004b:53015)
  • 15. W. H. Meeks III and H. Rosenberg.
    The uniqueness of the helicoid.
    Annals of Mathematics 161(2):723-754, 2005. MR 2153399
  • 16. W. H. Meeks III and H. Rosenberg
    The theory of minimal surfaces in $ M\times \mathbf{R}.$
    To appear in Comment. Math. Helv.
  • 17. B. Nelli and H. Rosenberg.
    Minimal surfaces in $ {\mathbf H}^2\times \mathbf{R}.$
    Bull. Bras. Math. Soc., New Series, 33(2):263-292, 2002. MR 1940353 (2004d:53014)
  • 18. B. Nelli and H. Rosenberg.
    Global Properties of Constant Mean Curvature Surfaces in $ \mathbf{H}^2\times \mathbf{R}$.
    To appear in Pacific J. of Mathematics.
  • 19. P. A. Nitsche.
    Existence of prescribed mean curvature graphs in hyperbolic space.
    Manuscripta Math., 108(3):349-367, 2002. MR 1918082 (2003f:53015)
  • 20. R. Pedrosa and M. Ritoré.
    Isoperimetric domains in the Riemannian product of a circle with simply connected space forms and applications to free boundary problems.
    Indiana Univ. J. of Mathematics, 48:1357-1394, 1999. MR 1757077 (2001k:53120)
  • 21. H. Rosenberg.
    Minimal surfaces in $ {M}^2\times {\mathbf {R}}$.
    Illinois J. Math, 46(4):1177-1195, 2002. MR 1988257 (2004d:53015)
  • 22. H. Rosenberg.
    Hypersurfaces of constant curvature in space forms.
    Bull Sc. Math, 117:211-239, 1993. MR 1216008 (94b:53097)
  • 23. R. Schoen.
    Analytic aspects of the harmonic map problem.
    In Seminar on Nonlinear Partial Differential Equations, 1983. MR 0765241 (86b:58032)
  • 24. J. Serrin.
    The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables.
    Philos. Trans. Roy. Soc. London Ser. A, 264:413-496, 1969. MR 0282058 (43:7772)
  • 25. J. Smoller.
    Shock waves and reaction-diffusion equations.
    Springer-Verlag, New York, 1983. MR 0688146 (84d:35002)
  • 26. M. Weber and M. Wolf.
    Teichmüller theory and handle addition for minimal surfaces.
    Annals of Math. 156:713-795, 2002. MR 1954234 (2005j:53012)
  • 27. M. Weber and M. Wolf.
    Minimal surfaces of least total curvature and moduli spaces of plane polygonal arcs.
    GAFA, 8:1129-1170, 1998. MR 1664793 (99m:53020)
  • 28. M. Wohlgemuth.
    Higher genus minimal surfaces by growing handles out of a catenoid.
    Manuscripta Math., 70:397-428, 1991. MR 1092145 (91k:53021)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 53C27, 58J60

Retrieve articles in all journals with MSC (2000): 53C27, 58J60

Additional Information

David Hoffman
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305

Jorge H. S. de Lira
Affiliation: Departamento de Matemática, Universidade Federal do Ceará, Fortaleza - Ceará - Brasil

Harold Rosenberg
Affiliation: Institut de Mathematiques de Jussieu, Paris XIII, France

Received by editor(s): December 1, 2003
Published electronically: September 26, 2005
Additional Notes: The first author was partially supported by research grant DE-FG03-95ER25250/A007 of the Applied Mathematical Science subprogram of the Office of Energy Research, U.S. Department of Energy, and National Science Foundation, Division of Mathematical Sciences research grant DMS-0139410.
The second author was partially supported by Cooperaçao Brasil-França - Ministère des Affaires ètrangères (France) and CAPES (Brasil)
Article copyright: © Copyright 2005 American Mathematical Society

American Mathematical Society