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Spacelike hypersurfaces with constant Gauss-Kronecker curvature in the Minkowski space. (English) Zbl 0828.53050
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.
Reviewer: F.Manhart (Wien)

##### MSC:
 53C40 Global submanifolds 53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
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##### References:
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