ON COMPLETE SPACELIKE HYPERSURFACES WITH CONSTANT m-TH MEAN CURVATURE IN AN ANTI-DE SITTER SPACE

Abstract

We investigate complete spacelike hypersurfaces in an Anti-de Sitter space with constant m-th mean curvature and two distinct principal curvatures. By using Otsuki's idea, we obtain some global classification results. For their application, we obtain some characterizations for hyperbolic cylinders. We prove that the only complete spacelike hypersurfaces in Anti-de Sitter (n + 1)-spaces (n ≥ 3) of constant mean curvature or constant scalar curvature with two distinct principal curvatures λ and µ satisfying inf(λ − µ) 2 > 0 are the hyperbolic cylinders. It is a little surprising that the corresponding result does not hold for m-th mean curvature when m > 2. We also obtain some global rigidity results for hyperbolic cylinders in terms of square length of the second fundamental form.

On complete spacelike hypersurfaces with two distinct principal curvatures in Lorentz–Minkowski space

“…The corresponding result of Theorem 5.1 for complete spacelike hypersurfaces in the anti-de Sitter space has been considered in [10,7]. Now let m ≤ n − 1, and M be a complete spacelike hypersurface in R n+1 1 (n ≥ 3) with constant mth mean curvature H m > 0 and two distinct principal curvatures λ and µ such that µ is simple and λ m > H m .…”

“…As for the case when the multiplicity of one of the two principal curvatures is n − 1, it corresponds to an ordinary differential equation. Otsuki's method can be generalized to study hypersurfaces in Riemannian space forms or spacelike hypersurfaces in Lorentzian space forms of constant mth mean curvature and two distinct principal curvatures (see e.g., [2][3][4][5][6][7]). …”

On spacelike hypersurfaces with two distinct principal curvatures in Lorentzian space forms

“…For example, by using Otsuki's idea [3], one can prove that the hyperbolic cylinders are the only complete spacelike hypersurfaces in anti-de Sitter space of constant mean curvature and two distinct principal curvatures λ and μ satisfying inf |λ − μ| > 0 [2,5,6]. Similar result also holds for spacelike hypersurfaces in Lorentz-Minkowski space [4].…”

On complete spacelike hypersurfaces with two distinct principal curvatures in a de Sitter space

“…We also obtain some global rigidity results for hyperbolic cylinders and obtain some non-existence results. Note that the corresponding global classification results for complete spacelike hypersurfaces with two distinct principal curvatures and constant m-th mean curvature in anti-de Sitter space and Lorentz-Minkowski space have been obtained recently by author [9,10].…”