On spacelike hypersurfaces with constant mean curvature in the de Sitter space
type: Item
instance of: scholarly article
title: On spacelike hypersurfaces with constant mean curvature in the de Sitter space
zbMATH Open document ID: 0611.53047
DOI: 10.1007/BF01179263
author: Kazuo Akutagawa
published in: Mathematische Zeitschrift
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173859
review text: In this paper, we shall prove the following Theorems 1 and 2 and show that our estimate (i) in Theorem 1 is optimal. Theorem 1. Let \(N=(N^{n+1}(c),g)\) be an \((n+1)\)-dimensional Lorentzian manifold of positive constant curvature c. Let M be a complete spacelike hypersurface with constant mean curvature H in N. Suppose that (i) \(| H| \leq c^{1/2}\) when \(n=2,\) (ii) \(| H| <2[(n-1)c]^{1/2}/n\) when \(n\geq 3.\) Then M is totally umbilical. Theorem 2. Let M be a compact spacelike surface with constant mean curvature in the 3-dimensional de Sitter space. Then M is totally umbilical. The fact that our estimate (i) in Theorem 1 is optimal is a consequence of the following. We can construct, for any constant H satisfying \(| H| >c^{1/2}\), complete noncompact spacelike surfaces embedded in the 3-dimensional de Sitter space of positive constant curvature c, which have constant mean curvature H and which are not totally umbilical.
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C50
zbMATH DE Number: 3988183
zbMATH Keywords: Lorentzian manifold
zbMATH Keywords: constant curvature
zbMATH Keywords: spacelike hypersurface
zbMATH Keywords: constant mean curvature
zbMATH Keywords: totally umbilical
zbMATH Keywords: de Sitter space
MaRDI profile type: Publication
cites work: Q5612020
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OpenAlex ID: W2073192231
description: scientific article; zbMATH DE number 3988183