Taiwanese Journal of Mathematics

$L_k$-2-TYPE HYPERSURFACES IN HYPERBOLIC SPACES

Pascual Lucas and Héctor-Fabián Ramírez-Ospina

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Abstract

In this article, we study $L_k$-finite-type hypersurfaces $M^n$ of a hyperbolic space $\mathbb{H}^{n+1}\subset\mathbb{R}^{n+2}_1$, for $k\geq 1$. In the 3-dimensional case, we obtain the following classification result. Let $\psi:M^3\rightarrow\mathbb{H}^{4}\subset\mathbb{R}^5_1$ be an orientable hypersurface with constant $k$-th mean curvature $H_k$, which is not totally umbilical. Then $M^3$ is of $L_k$-2-type if and only if $M^3$ is an open portion of a standard Riemannian product $\mathbb{H}^1(r_1)\times\mathbb{S}^{2}(r_2)$ or $\mathbb{H}^2(r_1)\times\mathbb{S}^{1}(r_2)$, with $-r_1^2+r_2^2=-1$. In the $n$-dimensional case, we show that a hypersurface $M^n\subset\mathbb{H}^{n+1}$, with constant $k$-th mean curvature $H_k$ and having at most two distinct principal curvatures, is of $L_k$-2-type if and only if $M^n$ is an open portion of a Riemannian product $\mathbb{H}^m(r_1)\times\mathbb{S}^{n-m}(r_2)$, with $-r_1^2+r_2^2=-1$, for some integer $m\in\{1,\dots,n-1\}$. In the case $k=n-1$ we drop the condition on the principal curvatures of the hypersurface $M^n$, and prove that if $M^n\subset\mathbb{H}^{n+1}$ is an orientable $H_{n-1}$-hypersurface of $L_{n-1}$-2-type then its Gauss-Kronecker curvature $H_{n}$ is a nonzero constant.

Article information

Source
Taiwanese J. Math., Volume 19, Number 1 (2015), 221-242.

Dates
First available in Project Euclid: 4 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499133627

Digital Object Identifier
doi:10.11650/tjm.19.2015.4603

Mathematical Reviews number (MathSciNet)
MR3313414

Zentralblatt MATH identifier
1357.53070

Subjects
Primary: 53C40: Global submanifolds [See also 53B25] 53B25: Local submanifolds [See also 53C40]

Keywords
hyperbolic hypersurface linearized operator $L_k$ $L_k$-finite-type hypersurface higher order mean curvatures Newton transformations

Citation

Lucas, Pascual; Ramírez-Ospina, Héctor-Fabián. $L_k$-2-TYPE HYPERSURFACES IN HYPERBOLIC SPACES. Taiwanese J. Math. 19 (2015), no. 1, 221--242. doi:10.11650/tjm.19.2015.4603. https://projecteuclid.org/euclid.twjm/1499133627


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