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

In this paper, we study complete hypersurfaces with constant mean curvature in anti-de Sitter space $H^{n+1}_1(-1)$. we prove that if a complete space-like hypersurface with constant mean curvature $x:\mathbf M\rightarrow H^{n+1}_1(-1)$ has two distinct principal curvatures $λ,μ$, and inf$|λ-μ|>0$, 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 $H^{n+1}_1 (-1)$.

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