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

TL;DR

This paper establishes existence, uniqueness, and classification of convex spacelike surfaces with prescribed curvature in (2+1)-dimensional Minkowski space, linking geometric PDEs to boundary function regularity and providing a foliation by constant curvature surfaces.

Contribution

It solves the Minkowski problem in Minkowski space for domains contained in the future cone, characterizes surfaces with bounded principal curvatures via Zygmund functions, and describes a foliation by constant curvature surfaces.

Findings

Existence and uniqueness of convex Cauchy surfaces with prescribed curvature.

Characterization of surfaces with bounded principal curvatures using Zygmund class functions.

Domains are foliated by constant curvature surfaces for all negative K.

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-Amp\`ere equation $\det D^2 u(z)=(1/\psi(z))(1-|z|^2)^{-2}$ on the unit disc, with the boundary condition $u|_{\partial\mathbb{D}}=\varphi$, for $\psi$ a smooth positive function and $\varphi$ 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 $\varphi$ 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 $\partial \mathbb{D}$. 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 $(-\infty,0)$.