Hausdorff-type metric geometry of the space of Cauchy hypersurfaces

TL;DR

This paper introduces a Hausdorff-type metric on the space of Cauchy hypersurfaces in globally hyperbolic spacetimes, analyzing its mathematical properties and generalizing existing completeness results in Lorentzian geometry.

Contribution

It defines a natural metric on Cauchy hypersurfaces and studies its properties, extending previous results on spacetime completeness to more general Lorentzian settings.

Findings

The space of Cauchy hypersurfaces is shown to be complete under the new metric.

The metric space is locally compact in certain Lorentzian contexts.

Generalization of completeness results by Beem and Takahashi to broader settings.

Abstract

We equip the space of Cauchy hypersurfaces in a globally hyperbolic spacetime with a natural Hausdorff-type metric and study its properties, in particular completeness and local compactness, for Lorentzian manifolds and in more general synthetic Lorentzian settings. For this purpose, we also generalize results on completeness properties of spacetimes due to Beem and Takahashi.