## Hokkaido Mathematical Journal

### Constant mean curvature spacelike hypersurfaces in standard static spaces: rigidity and parabolicity

#### Abstract

Our purpose in this paper is investigate the geometry of complete constant mean curvature spacelike hypersurfaces immersed in a standard static space, that is, a Lorentzian manifold endowed with a globally defined timelike Killing vector field. In this setting, supposing that the ambient space is a warped product of the type $M^n\times_{\rho}\mathbb{R}_1$ whose Riemannian base $M^n$ has nonnegative sectional curvature and the warping function $\rho$ is convex on $M^n$, we use the generalized maximum principle of Omori-Yau in order to establish rigidity results concerning these spacelike hypersurfaces. We also study the parabolicity of maximal spacelike surfaces in $M^2\times_{\rho}\mathbb{R}_1$ and we obtain uniqueness results for entire Killing graphs constructed over $M^n$.

#### Article information

Source
Hokkaido Math. J., Volume 49, Number 2 (2020), 297-323.

Dates
First available in Project Euclid: 7 October 2020

https://projecteuclid.org/euclid.hokmj/1602036027

Digital Object Identifier
doi:10.14492/hokmj/1602036027

Mathematical Reviews number (MathSciNet)
MR4159172

Zentralblatt MATH identifier
07276077

#### Citation

LIMA, Eudes L. de; LIMA, Henrique F. de; LIMA Jr., Eraldo A.; MEDEIROS, Adriano A. Constant mean curvature spacelike hypersurfaces in standard static spaces: rigidity and parabolicity. Hokkaido Math. J. 49 (2020), no. 2, 297--323. doi:10.14492/hokmj/1602036027. https://projecteuclid.org/euclid.hokmj/1602036027