Spacelike convex surfaces with prescribed curvature in (2 + 1)-Minkowski space

Abstract

We prove existence and uniqueness of solutions to the Minkowski problem in any domain of dependence D in (2+1)-dimensional Minkowski space, provided D is contained in the future cone over a point. Namely, it is possible to find a smooth convex Cauchy surface with prescribed curvature function on the image of the Gauss map. This is related to solutions of the Monge–Ampère equation det⁡D2u(z)=(1/ψ(z))(1−|z|2)−2 on the unit disc, with the boundary condition u|∂D=φ, for ψ a smooth positive function and φ a bounded lower semicontinuous function.We then prove that a domain of dependence D contains a convex Cauchy surface with principal curvatures bounded from below by a positive constant if and only if the corresponding function φ is in the Zygmund class. Moreover in this case the surface of constant curvature K contained in D has bounded principal curvatures, for every K<0. In this way we get a full classification of isometric immersions of the hyperbolic plane in Minkowski space with bounded shape operator in terms of Zygmund functions of ∂D.Finally, we prove that every domain of dependence as in the hypothesis of the Minkowski problem is foliated by the surfaces of constant curvature K, as K varies in (−∞,0).