Complete spacelike hypersurfaces with constant mean curvature in the de Sitter space: a gap theorem

Abstract

Let $M^n$ be a complete spacelike hypersurface with constant mean curvature $H$ in the de Sitter space $S_1^{n+1}$. We use the operator $\phi =A-HI$, where $A$ is the second fundamental form of $M$, and the roots $B_H^- \le B_H^+$ of a certain second order polynomial, to prove that either $\vert\phi\vert^2\equiv 0$ and $M$ is totally umbilical, or $B_H^-\le\sqrt{\sup \vert\phi\vert^2}\le B_H^+$. For the case $H\geq 2\sqrt{n-1}/n$ we prove the following results: for every number $B$ in the interval $[\max\{0,B_H^-\},B_H^+]$ there is an example of a complete spacelike hypersurface such that $\sqrt{\sup \vert\phi\vert^2}=B$; if $\sqrt{\sup \vert\phi\vert^2}=B_H^-$ is attained at some point, then the corresponding $M$ is a hyperbolic cylinder. We characterize the hyperbolic cylinders as the only complete spacelike hypersurfaces in $S_1^{n+1}$ with constant mean curvature, non-negative Ricci curvature and having at least two ends. We also characterize all complete spacelike hypersurfaces of constant mean curvature with two distinct principal curvatures as rotation hypersurfaces or generalized hyperbolic cylinders.