An Application of Maximum Principle to space-like Hypersurfaces with Constant Mean Curvature in Anti-de Sitter Space

Abstract

In this paper, we study complete hypersurfaces with constant mean curvature in anti-de Sitter space $H_1^{n+1}(-1)$. we prove that if a complete space-like hypersurface with constant mean curvature $x:\mathbf{M}\to H_1^{n+1}(-1)$ has two distinct principal curvatures $\lambda ,\mu$, and inf$|\lambda -\mu |>0$, then $x$ is the standard embedding $H^m(-\frac{1}{r^2})\times H^{n-m}(-\frac{1}{1-r^2})$in anti-de Sitter space $H_1^{n+1}(-1)$.

arXiv comment9 pages