Gromov hyperbolic domains in Minkowski space

TL;DR

This paper explores the properties of Gromov hyperbolic domains in Minkowski space, establishing new characterizations of hyperbolicity through boundary causality and comparing various intrinsic metrics.

Contribution

It provides novel equivalences between Gromov hyperbolicity and boundary acausality for convex domains, and compares multiple metrics within Minkowski space.

Findings

Gromov hyperbolicity is equivalent to boundary acausality in convex domains.

Established relationships between Markowitz, Sormani–Vega, and quasi-hyperbolic metrics.

Compared Markowitz metric with the Hilbert metric in Minkowski space.

Abstract

We investigate domains in Minkowski space that are Gromov hyperbolic with respect to a Kobayashi-like metric introduced by Markowitz in the 1980s. For convex, future complete domains, Gromov hyperbolicity is shown to be equivalent to the stable acausality of the boundary. An analogous characterization is obtained for bounded, convex, causally convex domains in terms of the stable acausality of their Geroch--Kronheimer--Penrose causal boundaries. Our approach is based on explicit comparisons between the Markowitz metric, the Sormani--Vega null distance and the quasi-hyperbolic metric. We also make use of dynamical arguments similar to those of Benoist and Zimmer in projective and complex geometry. Finally, we compare the Markowitz metric to the Hilbert metric.