The Gauss map of a spacelike constant mean curvature hypersurface of Minkowski space
The author studies analogues of the Bernstein problem for complete spacelike hypersurfaces in Minkowski space \(R^{n,1}\) with constant mean curvature. Let \(\eta\) be the timelike unit normal field to M, which can be viewed as the (harmonic) Gauss map from M to the hyperbolic space \(H^ n\) embedded as the unit ``sphere'' in \(R^{n,1}\). He proves the Gauss image must be large: Theorem I. Let \(M\subset R^{n,1}\) be a spacelike hypersurface with constant mean curvature \(H\neq 0\). There is \(\tau =\tau(n,H)>0\) such that if \(\eta(M)\) lies in a geodesic ball of radius \(\tau\) in \(H^ n\), then M is not complete. The proof relies on the following estimate for the first Dirichlet eigenvalue \(\lambda_ 1\) of a geodesic ball \(B_{\rho}\) of radius \(\rho\) in M: Theorem II. If \(B_{\rho}\subset M\), then \(\lambda_ 1(B_{\rho})\geq nH^ 2(\cosh \tau -1)^{-1}.\) Since there are upper bounds [\textit{M. Gage}, Indiana Univ. Math. J. 29, 897-912 (1980; Zbl 0465.53031)] for \(\lambda_ 1\) in terms of the minimum \(\underline R\) for the Ricci curvature on \(B_{\rho}\subset M\), and since \(\underline R\geq -n^ 2H^ 2/4\) [\textit{S. Y. Cheng} and \textit{S. T. Yau}, Ann. Math., II. Ser. 104, 407-419 (1976; Zbl 0352.53021)], one gets a contradiction if \(\tau\) is too small.
scientific article; zbMATH DE number 4151265