On the geometry of linear Weingarten spacelike hypersurfaces in the de Sitter space (Q354700)

The authors study linear Weingarten space-like hypersurfaces immersed in de Sitter space \(\mathbb S^{n+1}_1\). A linear Weingarten hypersurface is a hypersurface for which the mean curvature \(H\) and normalized scalar curvature \(R\) satisfy a relation \(R= aH+b\) for real constants \(a\) and \(b\). The authors assume that \((n-1)a^2 + 4n(1-b) \) is nonnegative in case the hypersurface \(M^n\) is compact and that \((n-1)a^2 + 4n(1-b) \) is strictly positive in case \(M^n\) is noncompact. Put briefly, the main results are as follows: (1) If \(M^n\) is complete and noncompact, \(H\) is bounded and \(|\nabla H|\) is Lebesgue integrable on \(M^n\), then \(M^n\) is either a totally umbilical hypersurface isometric to \(\mathbb R^n\) or \(M\) is isometric to a hyperbolic cylinder. (2) If \(M^n\) is complete and noncompact, \(|\nabla H|\) is Lebesgue integrable on \(M^n\), and the second fundamental form \(B\) satisfies \(\sup |B|^2 \leq 2\sqrt{n-1}\), then \(\sup |B|^2 =2\sqrt{n-1}\), and further, in case \(n=2\), \(M^2\) is a totally umbilical space-like surface isometric to \(\mathbb R^2\), while in case \(n\geq 3\), \(M^n\) is isometric to a hyperbolic or spherical cylinder (depending on the sign of \(R\)). (3) If \(M^n\) is compact and \(\sup |B|^2 \leq 2\sqrt{n-1}\), then \(M^n\) is a totally umbilical space-like hypersurface isometric, up to scaling, to \(\mathbb S^n\). The main analytical tool is an appropriate maximum principle for complete noncompact Riemannian manifolds.