On the regularity of Cauchy hypersurfaces and temporal functions in closed cone structures

TL;DR

This paper investigates the regularity and stability of Cauchy hypersurfaces and temporal functions within closed cone structures, establishing new regularity results and extension properties for spacelike hypersurfaces.

Contribution

It proves the stability of locally stably acausal Cauchy hypersurfaces, shows the regularity of signed distance functions, and demonstrates that Cauchy hypersurfaces can be characterized as level sets of regular Cauchy temporal functions.

Findings

Signed distance functions share regularity with hypersurfaces near them

Cauchy hypersurfaces are level sets of regular Cauchy temporal functions

Compact spacelike hypersurfaces with boundary can be extended to Cauchy hypersurfaces

Abstract

We complement our work on the causality of upper semi-continuous distributions of cones with some results on Cauchy hypersurfaces. We prove that every locally stably acausal Cauchy hypersurface is stable. Then we prove that the signed distance $d_S$ from a spacelike hypersurface $S$ is, in a neighborhood of it, as regular as the hypersurface, and by using this fact we give a proof that every Cauchy hypersurface is the level set of a Cauchy temporal (and steep) function of the same regularity as the hypersurface. We also show that in a globally hyperbolic closed cone structure compact spacelike hypersurfaces with boundary can be extended to Cauchy spacelike hypersurfaces of the same regularity. We end the work with a separation result and a density result.