Rigidity and nonexistence of complete spacelike hypersurfaces in the steady state space

TL;DR

This paper investigates the geometric properties of complete spacelike hypersurfaces in the steady state space, establishing conditions under which they must be hyperplanes and proving nonexistence results using advanced maximum principles.

Contribution

It provides new rigidity and nonexistence theorems for spacelike hypersurfaces in the steady state space under curvature constraints, extending maximum principle techniques.

Findings

Complete spacelike hypersurfaces under certain curvature conditions are hyperplanes.

Nonexistence of certain hypersurfaces in the steady state space.

Extension of the Omori-Yau maximum principle to this setting.

Abstract

We study complete spacelike hypersurfaces immersed in an open region of the de Sitter space $\mathbb{S}^{n+1}_{1}$ which is known as the steady state space $\mathcal{H}^{n+1}$. In this setting, under suitable constraints on the behavior of the higher order mean curvatures of these hypersurfaces, we prove that they must be spacelike hyperplanes of $\mathcal{H}^{n+1}$. Nonexistence results concerning these spacelike hypersurfaces are also given. Our approach is based on a suitable extension of the Omori-Yau's generalized maximum principle due to Al\'{\i}as, Impera and Rigoli in [5].