Taiwanese Journal of Mathematics

HYPERSURFACES IN SPACE FORMS SATISFYING THE CONDITION $L_k x = Ax + b$

Luis J. Alías and S. M. B. Kashani

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Abstract

We study hypersurfaces either in the sphere $\mathbb{S}^{n+1}$ or in the hyperbolic space $\mathbb{H}^{n+1}$ whose position vector $x$ satisfies the condition $L_k x = Ax + b$, where $L_k$ is the linearized operator of the $(k+1)$-th mean curvature of the hypersurface for a fixed $k = 0, \ldots, n−1$, $A \in \mathbb{R}^{(n+2) \times (n+2)}$ is a constant matrix and $b \in \mathbb{R}^{n+2}$ is a constant vector. For every $k$, we prove that when $A$ is self-adjoint and $b=0$, the only hypersurfaces satisfying that condition are hypersurfaces with zero $(k+1)$-th mean curvature and constant $k$-th mean curvature, and open pieces of standard Riemannian products of the form $\mathbb{S}^m(\sqrt{1-r^2}) \times \mathbb{S}^{n-m}(r) \subset \mathbb{S}^{n+1}$, with $0 \lt r \lt 1$, and $\mathbb{H}^m(−\sqrt{1+r^2}) \times \mathbb{S}^{n-m}(r) \subset \mathbb{H}^{n+1}$, with $r \gt 0$. If $H_k$ is constant, we also obtain a classification result for the case where $b \neq 0$.

Article information

Source
Taiwanese J. Math., Volume 14, Number 5 (2010), 1957-1977.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500406026

Digital Object Identifier
doi:10.11650/twjm/1500406026

Mathematical Reviews number (MathSciNet)
MR2724143

Zentralblatt MATH identifier
1221.53085

Citation

Alías, Luis J.; Kashani, S. M. B. HYPERSURFACES IN SPACE FORMS SATISFYING THE CONDITION $L_k x = Ax + b$. Taiwanese J. Math. 14 (2010), no. 5, 1957--1977. doi:10.11650/twjm/1500406026. https://projecteuclid.org/euclid.twjm/1500406026


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