Rigidity and nonexistence of complete spacelike hypersurfaces in the steady state space
Weiller F. C. Barboza, Henrique F. de Lima, Marco Antonio L. Velásquez
TL;DR
This work investigates complete spacelike hypersurfaces in the steady state space $\mathcal{H}^{n+1}$, establishing rigidity and nonexistence results under constraints on higher-order mean curvatures $H_r$. By extending Omori–Yau type maximum principles through Newton transformations $P_r$ and operators $L_r$, and exploiting height and angle functions $l_a$ and $f_a$, the authors show that under suitable curvature and geometric bounds such hypersurfaces must be spacelike hyperplanes $\mathcal{E}_{\tilde{\tau}}$. Key contributions include precise rigidity statements when $H>0$, $H\le H_2$, and $|a^\top|$ is suitably small; nonexistence results ruling out certain constant-$H_r$ configurations; and extended rigidity results for configurations with linear relations between height and angle functions as well as higher-order curvature controls. Collectively, the results advance Bernstein-type rigidity in cosmological steady-state models and illuminate the foliations by spacelike hyperplanes in $\mathcal{H}^{n+1}$.
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 Alías, Impera and Rigoli in [5].