Entire surfaces of constant curvature in Minkowski 3-space

TL;DR

This paper studies the global structure of properly embedded spacelike surfaces with constant Gaussian curvature in Minkowski 3-space, classifying such surfaces and exploring applications to hyperbolic geometry.

Contribution

It provides a classification of convex surfaces with prescribed Gaussian curvature in Minkowski space, extending the Minkowski problem and analyzing their foliations.

Findings

Regular domains not wedges are foliated by convex constant curvature surfaces.

Classification results for surfaces with bounded prescribed Gaussian curvature.

Applications to minimal Lagrangian self-maps of the hyperbolic plane.

Abstract

This paper concerns the global theory of properly embedded spacelike surfaces in three-dimensional Minkowski space in relation to their Gaussian curvature. We prove that every regular domain which is not a wedge is uniquely foliated by properly embedded convex surfaces of constant Gaussian curvature. This is a consequence of our classification of surfaces with bounded prescribed Gaussian curvature, sometimes called the Minkowski problem, for which partial results were obtained by Li, Guan-Jian-Schoen, and Bonsante-Seppi. Some applications to minimal Lagrangian self-maps of the hyperbolic plane are obtained.