Spacelike hypersurfaces with constant scalar curvature in the Lorentz-Minkowski space
type: Item
instance of: scholarly article
title: Spacelike hypersurfaces with constant scalar curvature in the Lorentz-Minkowski space
zbMATH Open document ID: 0949.53040
DOI: 10.1023/A:1006660924994
author: Luis J. Alías
author: José A. Pastor
published in: Annals of Global Analysis and Geometry
publication date: 2000-12-05T00:00:00Z
review text: It was shown by \textit{E. Calabi} [Proc. Sympos. Pure Math. 15, 223-230 (1970; Zbl 0211.12801)] and \textit{S.-Y. Cheng}, \textit{S.-T. Yau} [Ann. Math. 104, 407-419 (1976; Zbl 0352.53021)] that the only complete spacelike hypersurfaces in the Lorentz-Minkowski space \(L^{n+1}\) with zero mean curvature are the spacelike hyperplanes. It was also shown by \textit{R. Aiyama} [Tsukuba J. Math. 16, 353-361 (1992; Zbl 0782.53043)] and \textit{Y. L. Xin} [Comment. Math. Helv. 66, 590-598 (1991; Zbl 0752.53038)] that the only complete constant mean curvature spacelike hypersurfaces in \(L^{n+1}\) having the image of the Gauss map contained in a geodesic ball of hyperbolic \(n\)-space are the spacelike hyperplanes. In the paper under review, it is shown that the only compact spacelike hypersurfaces in \(L^{n+1}\) with nonzero constant scalar curvature and with spherical boundary are the hyperbolic caps with negative constant scalar curvature. The key ingredients for the proof are new integral formulas for the \(n\)-dimensional volume enclosed by the boundary of a compact spacelike hypersurface, in the case where the boundary is contained in a (necessarily spacelike) hyperplane of \(L^{n+1}\).
reviewed by: Sung-Eun Koh
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C50
zbMATH DE Number: 1422974
zbMATH Keywords: hyperbolic image
zbMATH Keywords: spacelike hypersurface
zbMATH Keywords: constant scalar curvature
zbMATH Keywords: hyperbolic cap
zbMATH Keywords: integral formula
MaRDI profile type: Publication
full work available at URL: a:1006660924994
OpenAlex ID: W2915054693
description: scientific article; zbMATH DE number 1422974