A note on invariant constant curvature immersions in Minkowski space

TL;DR

This paper proves the existence and uniqueness of certain convex surfaces in Minkowski space with prescribed constant curvature metrics, linking hyperbolic geometry, affine deformations, and globally hyperbolic spacetimes.

Contribution

It establishes a novel correspondence between constant curvature metrics on surfaces and invariant convex surfaces in Minkowski space via affine deformations of Fuchsian groups.

Findings

Existence and uniqueness of affine deformations for given metrics.

Construction of convex Cauchy surfaces in Minkowski quotients.

Duality with half-pipe spaces and realization as fundamental forms.

Abstract

Let $S$ be a compact, orientable surface of hyperbolic type. Let $(k_+,k_-)$ be a pair of negative numbers and let $(g_+, g_-)$ be a pair of marked metrics over $S$ of constant curvature equal to $k_+$ and $k_-$ respectively. Using a functional introduced by Bonsante, Mondello \& Schlenker, we show that there exists a unique affine deformation $\Gamma:=(\rho,\tau)$ of a Fuchsian group such that $(S,g_+)$ and $(S, g_-)$ embed isometrically as locally strictly convex Cauchy surfaces in the future and past complete components respectively of the quotient by $\Gamma$ of an open subset $\Omega$ of Minkowski space. Such quotients are known as Globally Hyperbolic, Maximal, Cauchy compact Min\-kow\-ski spacetimes and are naturally dual to the half-pipe spaces introduced by Danciger. When translated into this latter framework, our result states that there exists a unique, marked, quasi-Fuchsian half-pipe space in which $(S, g_+)$ and $(S, g_-)$ are realised as the third fundamental forms of future- and past-oriented, locally strictly convex graphs.