Complete spacelike hypersurfaces with constant mean curvature in \(-\mathbb R\times \mathbb H^n\)
type: Item
instance of: scholarly article
title: Complete spacelike hypersurfaces with constant mean curvature in \(-\mathbb R\times \mathbb H^n\)
zbMATH Open document ID: 1193.53124
published in: Journal of Mathematical Analysis and Applications
publication date: 2010-05-26T00:00:00Z
review text: By applying the Omori-Yau generalized maximum principle for complete Riemannian manifolds, the authors prove the following Bernstein-type results: Theorem 1.1. Let \(\psi:\Sigma^n\to-\mathbb{R}\times\mathbb{H}^n\) be a complete space-like hypersurface with constant mean curvature \(H\). If the height function \(h\) of \(\Sigma^n\) satisfies, for some constant \(0<\alpha<1\), \(\displaystyle|\nabla h|^2\leq\frac{n\alpha}{n-1}H^2\), then \(\Sigma^n\) is a slice. Theorem 1.2. Let \(\psi:\Sigma^n\to-\mathbb{R}\times\mathbb{H}^n\) be a complete space-like hypersurface with constant mean curvature \(H\), and the 2-mean curvature \(H_2\) bounded from below. If the height function \(h\) of \(\Sigma^n\) satisfies, for some constant \(0<\alpha<1\), \(\displaystyle |\nabla h|^2\leq\frac{\alpha}{n-1}|A|^2\), where \(|A|^2\) denotes the squared norm of the shape operator \(A\), then \(\Sigma^n\) is a slice. The first author has given some examples of complete and non-complete entire maximal graphs in \(-\mathbb{R}\times\mathbb{H}^2\) which are not slices [see, Differ. Geom. Appl. 26, No.~4, 456--462 (2008; Zbl 1147.53047)].
reviewed by: Weihuan Chen
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C50
zbMATH DE Number: 5712602
zbMATH Keywords: Lorentzian product space
zbMATH Keywords: spacelike hypersurface
zbMATH Keywords: mean curvature
author: Alma L. Albujer
author: Fernanda Ester Camillo Camargo
author: Henrique Fernandes de Lima
MaRDI profile type: Publication
full work available at URL: j.jmaa.2010.02.039
OpenAlex ID: W2077885017
cites work: New examples of entire maximal graphs in \(\mathbb H^2 \times \mathbb R_1\)
cites work: Calabi-Bernstein results for maximal surfaces in Lorentzian product spaces
cites work: Spacelike hypersurfaces with constant mean curvature in the steady state space
cites work: INTEGRAL FORMULAE FOR SPACELIKE HYPERSURFACES IN CONFORMALLY STATIONARY SPACETIMES AND APPLICATIONS
cites work: Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson–Walker spacetimes
cites work: Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson-Walker spacetimes
cites work: Complete vertical graphs with constant mean curvature in semi-Riemannian warped products
cites work: Isometric immersions of Riemannian manifolds
cites work: Q3312869
cites work: Harmonic functions on complete riemannian manifolds
DOI: 10.1016/J.JMAA.2010.02.039
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description: scientific article; zbMATH DE number 5712602