The paper deals with space-like convex hypersurfaces of positive constant (K-hypersurfaces) or prescribed Gauss curvature in Minkowski space \(\mathbb R_1^n\) where \(n \geq 2.\) More explicitly, the authors study the entire solutions on \(\mathbb R^n\) of the Monge-Ampère equation: \[ \text{ det} D^2 u = \psi(x,u)(1-| Du| ^2)^{\frac{n+2}{2}} \] with the space-like condition \[ | Du| <1, \] because each of these hypersurfaces can be expressed as the graph of a convex function \(x_{n+1}=u(x)\) satisfying the Monge-Ampère equation and the space-like condition where \(\psi\) is the Gauss curvature.
The authors provide a classification for entire space-like K-hypersurfaces admitting a rotational symmetry with respect to a space-like axis. They are also interested in determining entire K-hypersurfaces in Minkowski space with bounded Gauss curvature admitting a prescribed tangent cone at infinity.
Moreover, the Minkowski type problem is considered due to its close relation to the problem of finding entire space-like hypersurfaces with prescribed Gauss curvature and extensions of some of the results in [A.-M. Li, Arch. Math. 64, No. 6, 534–551 (1995; Zbl 0828.53050)] are obtained.