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A new characterization of geodesic spheres in the hyperbolic space

Author: Jie Wu
Journal: Proc. Amer. Math. Soc. 144 (2016), 3077-3084
MSC (2010): Primary 53C24; Secondary 53C42
Published electronically: March 18, 2016
MathSciNet review: 3487237
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Abstract: This paper gives a new characterization of geodesic spheres in the hyperbolic space in terms of a ``weighted'' higher order mean curvature. Precisely, we show that a compact hypersurface $ \Sigma ^{n-1}$ embedded in $ \mathbb{H}^n$ with $ VH_k$ being constant for some $ k=1,\cdots ,n-1$ is a centered geodesic sphere. Here $ H_k$ is the $ k$-th normalized mean curvature of $ \Sigma $ induced from $ \mathbb{H}^n$ and $ V=\cosh r$, where $ r$ is a hyperbolic distance to a fixed point in $ \mathbb{H}^n$. Moreover, this result can be generalized to a compact hypersurface $ \Sigma $ embedded in $ \mathbb{H}^n$ with the ratio $ V\left (\frac {H_k}{H_j}\right )\equiv$$ \mbox {constant},\;0\leq j< k\leq n-1$ and $ H_j$ not vanishing on $ \Sigma $.

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  • [1] Luis J. Alías, Debora Impera, and Marco Rigoli, Hypersurfaces of constant higher order mean curvature in warped products, Trans. Amer. Math. Soc. 365 (2013), no. 2, 591-621. MR 2995367,
  • [2] Juan A. Aledo, Luis J. Alías, and Alfonso Romero, Integral formulas for compact space-like hypersurfaces in de Sitter space: applications to the case of constant higher order mean curvature, J. Geom. Phys. 31 (1999), no. 2-3, 195-208. MR 1706636 (2000e:53092),
  • [3] A. D. Aleksandrov, Uniqueness theorems for surfaces in the large. II, Vestnik Leningrad. Univ. 12 (1957), no. 7, 15-44 (Russian, with English summary). MR 0102111 (21 #906)
  • [4] Luis J. Alías, Jorge H. S. de Lira, and J. Miguel Malacarne, Constant higher-order mean curvature hypersurfaces in Riemannian spaces, J. Inst. Math. Jussieu 5 (2006), no. 4, 527-562. MR 2261223 (2007i:53062),
  • [5] Luis J. Alías and J. Miguel Malacarne, Constant scalar curvature hypersurfaces with spherical boundary in Euclidean space, Rev. Mat. Iberoamericana 18 (2002), no. 2, 431-442. MR 1949835 (2004b:53101),
  • [6] João Lucas Marques Barbosa and Antônio Gervasio Colares, Stability of hypersurfaces with constant $ r$-mean curvature, Ann. Global Anal. Geom. 15 (1997), no. 3, 277-297. MR 1456513 (98h:53091),
  • [7] Irl Bivens, Integral formulas and hyperspheres in a simply connected space form, Proc. Amer. Math. Soc. 88 (1983), no. 1, 113-118. MR 691289 (84k:53052),
  • [8] S. Brendle, Constant mean curvature surfaces in warped product manifolds, arXiv:1105.4273, to appear in Publ. Math. IHES.
  • [9] S. Brendle and M. Eichmair, Isoperimetric and Weingarten surfaces in the Schwarzschild manifold, arXiv:1208:3988, to appear in J. Diff. Geom.
  • [10] S. Brendle, P.-K. Hung, and M.-T. Wang, A Minkowski-type inequality for hypersurfaces in the Anti-deSitter-Schwarzschild manifold, arXiv: 1209.0669.
  • [11] M. Dahl, R. Gicquaud and A. Sakovich, Penrose type inequalities for asymptotically hyperbolic graphs, Annales Henri Poincaré (2012), DOI 10.1007/s00023-012-0218-4.
  • [12] L.L. de Lima and F. Girão, An Alexandrov-Fenchel-type inequality in hyperbolic space with an application to a Penrose inequality, arXiv:1209.0438v2.
  • [13] Lars Gårding, An inequality for hyperbolic polynomials, J. Math. Mech. 8 (1959), 957-965. MR 0113978 (22 #4809)
  • [14] Y. Ge, G. Wang and J. Wu, Hyperbolic Alexandrov-Fenchel quermassintegral inequalities I, arXiv:1303.1714.
  • [15] Y. Ge, G. Wang and J. Wu, Hyperbolic Alexandrov-Fenchel quermassintegral inequalities II, arXiv:1304.1417.
  • [16] Y. Ge, G. Wang and J. Wu, The GBC mass for asymptotically hyperbolic manifolds, preprint.
  • [17] P. Guan, Topics in Geometric Fully Nonlinear Equations, Lecture Notes,
  • [18] Yijun He, Haizhong Li, Hui Ma, and Jianquan Ge, Compact embedded hypersurfaces with constant higher order anisotropic mean curvatures, Indiana Univ. Math. J. 58 (2009), no. 2, 853-868. MR 2514391 (2010i:53109),
  • [19] Oussama Hijazi, Sebastián Montiel, and Xiao Zhang, Dirac operator on embedded hypersurfaces, Math. Res. Lett. 8 (2001), no. 1-2, 195-208. MR 1825270 (2002j:53053),
  • [20] Sung-Eun Koh, A characterization of round spheres, Proc. Amer. Math. Soc. 126 (1998), no. 12, 3657-3660. MR 1469418 (99b:53083),
  • [21] Sung-Eun Koh, Sphere theorem by means of the ratio of mean curvature functions, Glasg. Math. J. 42 (2000), no. 1, 91-95. MR 1739688 (2001a:53087),
  • [22] N. J. Korevaar, Sphere theorems via Alexsandrov for constant Weingarten curvature hypersurfaces--Appendix to a note of A. Ros, J. Diff. Geom. 27, (1988).
  • [23] H. Li, Y. Wei and C. Xiong, A geometric ineqality on hypersurface in hyperbolic space, arXiv:1211.4109.
  • [24] H. Ma and C. Xiong, Hypersurfaces with constant anisotropic mean curvatures, arXiv:1302:2992.
  • [25] Sebastián Montiel, Unicity of constant mean curvature hypersurfaces in some Riemannian manifolds, Indiana Univ. Math. J. 48 (1999), no. 2, 711-748. MR 1722814 (2001f:53131),
  • [26] Sebastián Montiel and Antonio Ros, Compact hypersurfaces: the Alexandrov theorem for higher order mean curvatures, Differential geometry, Pitman Monogr. Surveys Pure Appl. Math., vol. 52, Longman Sci. Tech., Harlow, 1991, pp. 279-296. MR 1173047 (93h:53062)
  • [27] Robert C. Reilly, Applications of the Hessian operator in a Riemannian manifold, Indiana Univ. Math. J. 26 (1977), no. 3, 459-472. MR 0474149 (57 #13799)
  • [28] Antonio Ros, Compact hypersurfaces with constant scalar curvature and a congruence theorem, J. Differential Geom. 27 (1988), no. 2, 215-223. With an appendix by Nicholas J. Korevaar. MR 925120 (89b:53096)
  • [29] Antonio Ros, Compact hypersurfaces with constant higher order mean curvatures, Rev. Mat. Iberoamericana 3 (1987), no. 3-4, 447-453. MR 996826 (90c:53160),
  • [30] Harold Rosenberg, Hypersurfaces of constant curvature in space forms, Bull. Sci. Math. 117 (1993), no. 2, 211-239. MR 1216008 (94b:53097)
  • [31] G. Wang and C. Xia, Isoperimetric type problems and Alexandrov-Fenchel type inequalities in the hyperbolic space, arXiv:1304.1674.

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Additional Information

Jie Wu
Affiliation: School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, People’s Republic of China; and Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Eckerstr. 1, D-79104 Freiburg, Germany
Address at time of publication: Department of Mathematics, Zhejiang University, Hangzhou 310027, People’s Republic of China

Received by editor(s): May 13, 2013
Published electronically: March 18, 2016
Additional Notes: The author was partly supported by SFB/TR71 “Geometric partial differential equations” of DFG; and by NSF of China under grant no. 11401553.
Communicated by: Lei Ni
Article copyright: © Copyright 2016 American Mathematical Society

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