## Abstract

**1. Introduction**

In this paper we are concerned with spacelike convex hypersurfaces of positive constant (Κ-hypersurfaces) or prescribed Gauss curvature in Minkowski space ℝ^{n, 1} (*n* ≧ 2). Any such hypersurface may be written locally as the graph of a convex function *x*
_{n+1} = *u*(*x*), *x* ε ℝ^{n} satisfying the spacelike condition

(1.1) |*Du*| < 1

and the Monge-Ampère type equation

(1.2)

where ψ is a prescribed positive function (the Gauss curvature). Our main purpose is to study entire solutions on ℝ^{n} of (1.1)–(1.2).

For ψ = 1 a well known entire solution of (1.1)–(1.2) is the hyperboloid

(1.3)

which gives an isometric embedding of the hyperbolic space ℍ^{n} into ℝ^{n, 1}. Hano and Nomizu [*J. Hano* and *K. Nomizu*, On isometric immersions of the hyperbolic plane into the Lorentz-Minkowski space and the Monge-Ampère equation of a certain type, Math. Ann. **262** (1983), 245–253.] were probably the first to observe the non-uniqueness of isometric embeddings of ℍ^{2} in ℝ^{2, 1} by constructing other (geometrically distinct) entire solutions of (1.1)–(1.2) for *n* = 2 (and ψ ≡ 1) using methods of ordinary differential equations. Using the theory of Monge-Ampère equations, A.-M. Li [*A.-M. Li*, Spacelike hypersurfaces with constant Gauss-Kronecker curvature in the Minkowski space, Arch. Math. **64** (1995), 534–551.] studied entire spacelike Κ-hypersurfaces with uniformly bounded principal curvatures, while the Dirichlet problem for (1.1)–(1.2) in a bounded domain Ω ⊂ ℝ^{n} was treated by Delanoë [*Ph. Delanoë*, The Dirichlet problem for an equation of given Lorentz-Gaussian curvature, Ukrainian Math. J. **42** (1990), 1538–1545.] when Ω is strictly convex, and by Guan [*B. Guan*, The Dirichlet problem for Monge-Ampère equations in non-convex domains and spacelike hyper-surfaces of constant Gauss curvature, Trans. Amer. Math. Soc. **350** (1998), 4955–4971.] for general (non-convex) Ω. In this paper we are interested in entire spacelike Κ-hypersurfaces, and more generally hypersurfaces of prescribed Gauss curvature, without a boundedness assumption on principal curvatures.

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