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

TL;DR

This paper applies the maximum principle to classify complete space-like hypersurfaces with constant mean curvature in anti-de Sitter space, showing they are standard embeddings under certain curvature conditions.

Contribution

It proves a classification theorem for hypersurfaces with two distinct principal curvatures in anti-de Sitter space, extending geometric understanding of such hypersurfaces.

Findings

Hypersurfaces with two distinct principal curvatures are standard embeddings.

The classification holds under the condition that the difference of principal curvatures is bounded away from zero.

The results apply to complete 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 $\lambda,\mu$, 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^{n+1}_1 (-1)$.