Completeness of convex entire surfaces in Minkowski 3-space

TL;DR

This paper investigates the conditions under which convex entire surfaces in Minkowski 3-space are complete or incomplete, based on their null support functions, contributing to the understanding of isometric embeddings of hyperbolic planes.

Contribution

It provides new criteria linking null support functions to the completeness of entire surfaces in Minkowski space, extending to surfaces with bounded curvature.

Findings

Complete surfaces correspond to sufficiently tame null support functions.

Incomplete surfaces arise from sufficiently sharp null support functions.

Results apply to surfaces with merely bounded curvature.

Abstract

We prove four results towards a description, in terms of the null support function, of the set of isometric embeddings of the hyperbolic plane into Minkowski 3-space. We show that for sufficiently tame null support function, the corresponding entire surface of constant curvature -1 is complete, and for sufficiently sharp null support function, it is incomplete. Our results apply also to entire surfaces whose curvature is merely bounded.