Hokkaido Mathematical Journal

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

Eudes L. de LIMA, Henrique F. de LIMA, Eraldo A. LIMA Jr., and Adriano A. MEDEIROS

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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

Permanent link to this document
https://projecteuclid.org/euclid.hokmj/1602036027

Digital Object Identifier
doi:10.14492/hokmj/1602036027

Mathematical Reviews number (MathSciNet)
MR4159172

Zentralblatt MATH identifier
07276077

Subjects
Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]
Secondary: 53B30: Lorentz metrics, indefinite metrics 53C50: Lorentz manifolds, manifolds with indefinite metrics

Keywords
Standard static spaces complete spacelike hypersurfaces constant mean curvature maximal spacelike surfaces entire Killing graphs

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


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