Shichang Shu and Annie Yi Han: Hypersurfaces in a hyperbolic space with constant $k$-th mean curvature, p.65-78

Abstract:

Let $M^n$ be an $n$-dimensional complete connected and oriented hypersurface in a hyperbolic space $H^{n+1}(c)$ with constant $k$-th mean curvature $H_k>0(k<n)$ and with two distinct principal curvatures, one of which is simple. In this paper, we show that $M^n$ is isometric to the Riemannian product $H^1(c_1)\times S^{n-1}(c_2)$ or $H^{n-1}(c_1)\times S^1(c_2)$, $\frac{1}{c_1}+\frac{1}{c_2}=\frac{1}{c}$, $c_1<0$, $c_2>0$ if $S\geq (n-1)t_2^2+c^2t_2^{-2}$ on $M^n$, or $S\leq(n-1)t_2^2+c^2t_2^{-2}$ on $M^n$, where $t_2$ is the positive real root of (1.6). We extend recent result of Z.Hu et al. [6].

Key Words: Complete hypersurface, $k$-th mean curvature, principal curvature.

2000 Mathematics Subject Classification: Primary: 53C42,
Secondary: 53A10.

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