Spacelike hypersurfaces with constant scalar curvature in the Lorentz-Minkowski space
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 \(L^{n+1}\).
scientific article; zbMATH DE number 1422974