Advertisement

Hypersurfaces with nonegative Ricci curvature in \(\mathbb {H}^{n+1}\)

  • Vincent Bonini
  • Shiguang MaEmail author
  • Jie Qing
Article
  • 53 Downloads

Abstract

Based on properties of n-subharmonic functions we show that a complete, noncompact, properly embedded hypersurface with nonnegative Ricci curvature in hyperbolic space has an asymptotic boundary at infinity of at most two points. Moreover, the presence of two points in the asymptotic boundary is a rigidity condition that forces the hypersurface to be an equidistant hypersurface about a geodesic line in hyperbolic space. This gives an affirmative answer to the question raised by Alexander and Currier (Proc Symp Pure Math 54(3):37–44, 1993).

Mathematics Subject Classification

53C40 53C21 

Notes

References

  1. 1.
    Alexander, S., Currier, R.J.: Nonnegatively curved hypersurfaces of hyperbolic space and subharmonic functions. J. Lond. Math. Soc. 41(2), 347–360 (1990)CrossRefGoogle Scholar
  2. 2.
    Alexander, S., Currier, R.J.: Hypersurfaces and nonnegative curvature. Proc. Symp. Pure Math. 54(3), 37–44 (1993)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Besse, A.L.: Einstein Manifolds. Springer, Berlin (1987)CrossRefGoogle Scholar
  4. 4.
    Bonini, V., Ma, S., Qing, J.: On nonnegatively curved hypersurfaces in hyperbolic space. Math. Ann. (2018).  https://doi.org/10.1007/s00208-018-1694-8
  5. 5.
    Bourguignon, J.P.: Les variétés de dimension 4 à signature non nulle dont la courbure est harmonique sont d’Einstein. Invent. Math. 63, 263–286 (1981)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cartan, E.: Familles de surfaces isoparamétriques dans les espaces à courbure constante. Ann. Mat. Pura Appl. (4) 17, 177–191 (1938)CrossRefGoogle Scholar
  7. 7.
    Cheeger, J., Gromoll, D.: The splitting theorem for manifolds of nonnegative Ricci curvature. J. Differ. Geom. 6, 119–128 (1971)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Currier, R.J.: On surfaces of hyperbolic space infinitesimally supported by horospheres. Trans. Am. Math. Soc. 313(1), 419–431 (1989)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Epstein, C.L.: Envelopes of horospheres and Weingarten Surfaces in Hyperbolic 3-Space, Unpublished (1986). http://www.math.upenn.edu/~cle/papers/index.html
  10. 10.
    Epstein, C.L.: The asymptotic boundary of a surface imbedded in \(\mathbb{H} ^3\) with nonnegative curvature. Michigan Math. J. 34, 227–239 (1987)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Heinonen, J., Kilpelainen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford University Press, Oxford (1993)zbMATHGoogle Scholar
  12. 12.
    Lindqvist, P.: Notes on the p-Laplace Equation. University of Jyvaskyla Lecture Notes (2006)Google Scholar
  13. 13.
    Toponogov, V.A.: Riemannian spaces which contain straight lines. Am. Math. Soc. Transl. 37(2), 287–290 (1964)zbMATHGoogle Scholar
  14. 14.
    Shen, Z., Sormani, C.: The topology of open manifolds with nonnegative Ricci curvature. Commun. Math. Anal. Conference 1, 20–34 (2008)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Volkov, Yu.A., Vladimirova, S.M.: Isometric immersions in the Euclidean plane in Lobachevskii space. Math. Zametki 10, 327–332 (1971); Math. Notes 10 (1971), 619–622Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsCal Poly State UniversitySan Luis ObispoUSA
  2. 2.Department of Mathematics and LPMCNankai UniversityTianjinChina
  3. 3.Department of MathematicsUniversity of CaliforniaSanta CruzUSA

Personalised recommendations