$L_k$-2-TYPE HYPERSURFACES IN HYPERBOLIC SPACES

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.