Spacelike hypersurfaces in de Sitter space with constant higher-order mean curvature
Let \(\mathbb{R}_{1}^{n+2}\) be a real vector space \(\mathbb{R}^{n+2}\) endowed with the Lorentzian metric \(<.,.>\) given by \(<x,y>=-x_{0}y_{0}+\sum_{i=1}^{n+2}x_{i}y_{i}\), for \(x\),\(y\) \(\in\mathbb{R}^{n+2}\). The de Sitter space can be defined as \(\mathbb{S}_{1}^{n+1}(c)=\{x\in \mathbb{R}_{1}^{n+2}| | x| ^{2}=\frac{1}{c}, \frac{1}{c}, > 0\}\) and inherits from \(<.,.>\) a metric which makes it an indefinite Riemannian manifold of constant sectional curvature \(c\). Let \(M\) be a space-like hypersurface immersed in the de Sitter space \(\mathbb{S}_{1}^{n+1}(c)\) and let \(k_{i}\) and \(e_{i}\), \(i=1,\dots,n\) be the principal curvatures and principal directions at a point \(p\). The \(r\)th mean curvature is defined by the identity \(P_{n}(t)= (1+tk_{1})\dots (1+tk_{n})=1+(_{1}^{n})H_{1}t+(_{n}^{n})H_{1}t^{n}\) for all \(t\in \mathbb{R}\), thus \(H_{1}=H\) is the mean curvature, \(H_{2}=\frac{n^{2}H^{2}-S}{n(n-1)}\), where \(S\) is the square length of the second fundamental form and \(H_{n}\) is the Gauss Kronecker curvature of \(M\) immersed in \(\mathbb{S}_{1}^{n+1}(c)\). By establishing a generalized Minkowski formula a generalization of results of \textit{S. Montiel} (for \(r=1\)) [Indiana Univ. Math. J. 37, No.4, 909-917 (1988; Zbl 0677.53067)] and \textit{Q.-M. Cheng} and \textit{S. Ishikwa} (for \(r=2\)) [Manuscr. Math. 95, No.4, 499-505 (1998; Zbl 0913.53022)] is given by showing that a compact connected hypersurface with positive constant higher order mean curvature \(H_{r}\) for some fixed \(r\), \( 1\leq r \leq n\) immersed in the de Sitter space \(\mathbb{S}_{1}^{n+1}(c)\) must be a sphere.
scientific article; zbMATH DE number 5164006
scientific article