Spacelike hypersurfaces with constant Gauss-Kronecker curvature in the Minkowski space
It is well known that there is a standard embedding of the hyperbolic space \(H^2 (-1)\) into the Minkowski space \(\mathbb{R}^3_1\) as the spacelike surface \[ x^2_1 + x^2_2 - x^2_3 = -1\quad (x_3 > 0). \] Denoting the Gauss-Kronecker curvature with \(K\), the intrinsic Gauss-curvature (with respect to the Minkowski inner product) is \(-K\). The present paper deals with the problem of classifying the complete, spacelike, convex hypersurfaces with constant Gauss-Kronecker curvature \(K = 1\); the author restricts the discussion to the case of bounded principal curvatures. The main theorem says that, on the one hand, for every hypersurface in \(\mathbb{R}^{n + 1}_1\) of the above type there exists a function \(\varphi \in C^\infty (\partial B^n(1))\) \((B^n(1)\) the unit Ball), solving a certain Monge-Ampere equation, and conversely, given such a \(\varphi \in C^\infty\) \((\partial B^n (1))\) one can construct a hypersurface of the above type.
scientific article; zbMATH DE number 769541