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

Calculus of Variations and Partial Differential Equations volume 58, Article number: 36 (2019) Cite this article

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).

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References

  1. Alexander, S., Currier, R.J.: Nonnegatively curved hypersurfaces of hyperbolic space and subharmonic functions. J. Lond. Math. Soc. 41(2), 347–360 (1990)

    Article  Google Scholar 

  2. Alexander, S., Currier, R.J.: Hypersurfaces and nonnegative curvature. Proc. Symp. Pure Math. 54(3), 37–44 (1993)

    Article  MathSciNet  Google Scholar 

  3. Besse, A.L.: Einstein Manifolds. Springer, Berlin (1987)

    Book  Google Scholar 

  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. 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)

    Article  MathSciNet  Google Scholar 

  6. Cartan, E.: Familles de surfaces isoparamétriques dans les espaces à courbure constante. Ann. Mat. Pura Appl. (4) 17, 177–191 (1938)

    Article  Google Scholar 

  7. Cheeger, J., Gromoll, D.: The splitting theorem for manifolds of nonnegative Ricci curvature. J. Differ. Geom. 6, 119–128 (1971)

    Article  MathSciNet  Google Scholar 

  8. Currier, R.J.: On surfaces of hyperbolic space infinitesimally supported by horospheres. Trans. Am. Math. Soc. 313(1), 419–431 (1989)

    Article  MathSciNet  Google Scholar 

  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 Epstein, C.L.: Envelopes of horospheres and Weingarten Surfaces in Hyperbolic 3-Space, Unpublished (1986).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. Epstein, C.L.: The asymptotic boundary of a surface imbedded in \(\mathbb{H} ^3\) with nonnegative curvature. Michigan Math. J. 34, 227–239 (1987)

    Article  MathSciNet  Google Scholar 

  11. Heinonen, J., Kilpelainen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford University Press, Oxford (1993)

    MATH  Google Scholar 

  12. Lindqvist, P.: Notes on the p-Laplace Equation. University of Jyvaskyla Lecture Notes (2006)

  13. Toponogov, V.A.: Riemannian spaces which contain straight lines. Am. Math. Soc. Transl. 37(2), 287–290 (1964)

    MATH  Google Scholar 

  14. Shen, Z., Sormani, C.: The topology of open manifolds with nonnegative Ricci curvature. Commun. Math. Anal. Conference 1, 20–34 (2008)

    MathSciNet  MATH  Google Scholar 

  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), 619–622

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Authors and Affiliations

  1. Department of Mathematics, Cal Poly State University, San Luis Obispo, CA, 93407, USA

    Vincent Bonini

  2. Department of Mathematics and LPMC, Nankai University, Tianjin, China

    Shiguang Ma

  3. Department of Mathematics, University of California, Santa Cruz, CA, 95064, USA

    Jie Qing

Authors
  1. Vincent Bonini
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  2. Shiguang Ma
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  3. Jie Qing
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Corresponding author

Correspondence to Shiguang Ma.

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Communicated by A. Chang.

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Shiguang Ma is partially supported by NSFC 11571185 and 11871283

Jie Qing is partially supported by NSF DMS-1608782.

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Bonini, V., Ma, S. & Qing, J. Hypersurfaces with nonegative Ricci curvature in \(\mathbb {H}^{n+1}\). Calc. Var. 58, 36 (2019). https://doi.org/10.1007/s00526-018-1471-2

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  • DOI: https://doi.org/10.1007/s00526-018-1471-2

Mathematics Subject Classification

  • 53C40
  • 53C21