Gromov hyperbolic domains in Minkowski space

TL;DR

The paper identifies precise geometric conditions under which Markowitz’s Markowitz pseudodistance δ_Ω on a domain Ω in Minkowski space yields a Gromov hyperbolic metric space. By linking δ_Ω with the quasi-hyperbolic metric k_Ω and with the null distance hat{d}_τ, the authors reduce hyperbolicity to stable acausality properties of the causal boundary ∂_cΩ, establishing equivalences in bounded, convex, causally convex settings, and extending to future-complete unbounded domains under appropriate boundary conditions. The work also develops a robust comparison framework with the null distance, showing quasi-isometric relations in key regimes and translating the hyperbolicity results to causal boundary and cosmological time structures. It further demonstrates the limits of these results via examples (e.g., spacelike slabs, Bonsante domains), analyzes maximal domains, and contrasts Markowitz’s metric with the Hilbert metric, highlighting when they are not quasi-isometric. Overall, the paper provides a comprehensive Lorentzian analogue of classical Hilbert/Kobayashi hyperbolicity results, linking boundary causality to coarse geometric properties and offering tools for analyzing global hyperbolicity via δ_Ω and related metrics.

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.

Figures (7)

• Figure 1: Argument in the proof of Proposition \ref{['prop_estimee']}.
• Figure 2: Argument in the proof of Theorem \ref{['thm_equivalence_d_lnt_delta_Omega']}.
• Figure 3: A non-thin family of causal geodesic triangles of $(\Omega,\delta_\Omega)$, inside a future complete domain $\Omega$, whose boundary contains a lightlike half-line (in blue).
• Figure 4: A non-thin family of causal geodesic triangles of $(\Omega,\delta_\Omega)$, inside a bounded causally convex domain $\Omega$ such that $\partial\Omega$ contains a broken lightlike segment (in blue).
• Figure 5: The "zooming" step in the proof of Lemma \ref{['lemme_dynamique_cas_causalement_convexe']}. We depict the effect of the dilation and the isometry part of the similarity $h_k$ on $\Omega$.

Theorems & Definitions (121)

• Theorem 1.2: Benoist
• Theorem 1.3: Zimmer
• Theorem A
• Theorem B
• Remark 1.4
• Theorem C
• Theorem D
• Definition 2.1
• Proposition 2.2
• Definition 2.3