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 \({H^{n+1}_1(-1)}\). 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 \({ H^{n+1}_1 (-1) }\).