arXiv subjects

Adam Chalumeau

Publications and source records attributed to Adam Chalumeau.

3 recordsLinked to original sources

arXiv2026

Gromov hyperbolic domains in Minkowski space

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.

math.DG↗

arXiv2025

On Markowitz's pseudodistance for conformal manifolds

In the 1980s, M. J. Markowitz introduced a conformally invariant pseudodistance on pseudo-Riemannian manifolds, inspired by the Kobayashi metric in projective geometry. This construction relies on a distinguished class of parametrized lightlike geodesics, called projectively parametrized. We begin by reviewing the fundamental properties of this pseudodistance and provide several families of examples where it is non-degenerate and, in some cases, complete. In particular, we investigate three classes of manifolds: closed manifolds, conformally convex domains of the Einstein universe, and globally hyperbolic, conformally flat, $C$-maximal spacetimes. For the first two classes, we obtain results analogous to those of Brody and Barth concerning the complex Kobayashi metric. Finally, we apply Markowitz's pseudodistance to classify all quasi-homogeneous domains of the Einstein-de Sitter space, that is, a half-space of Minkowski space bounded by a spacelike hyperplane. Up to conformal transformations, only finitely many such domains exist, and all of them turn out to be homogeneous.

math.DG↗

arXiv2024

Proper Almost-Homogeneous Domains of the Einstein Universe

The Einstein universe $\mathbf{Ein}^{p,q}$ of signature $(p,q)$ is a pseudo-Riemannian analogue of the conformal sphere; it is the conformal compactification of the pseudo-Riemannian Minkowski space. For $p,q \geq 1$, we show that, up to a conformal transformation, there is only one almost-homogeneous domain in $\mathbf{Ein}^{p,q}$ that is bounded in a suitable stereographic projection. This domain, which we call a diamond, is a model for the symmetric space of $\operatorname{PO}(p,1) \times \operatorname{PO}(1,q)$. We deduce a classification of closed conformally flat manifolds with proper development.

math.DG↗