Entire spacelike hypersurfaces of prescribed Gauss curvature in Minkowski space

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 ℝⁿ 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 ℝⁿ 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 ℍⁿ 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 ℍ² 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 ⊂ ℝⁿ 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.