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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
DOI: https://doi.org/10.1090/proc/12325
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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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
Email: wujiewj@zju.edu.cn

DOI: https://doi.org/10.1090/proc/12325
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