Work

Polyhedral surfaces in flat (2 + 1)-spacetimes and balanced cellulations on hyperbolic surfaces

Year: 2025

Type: article

Abstract: We first prove that, given a hyperbolic metric ℎ on a closed surface 𝑆, any flat metric on 𝑆 with negative singular curvatures isometrically embeds as a convex polyhedral Cauchy surface in a unique future-complete flat globally hyperbolic maximal ( 2 + 1 ) (2+1) -spacetime whose linear part of the holonomy is given by ℎ. The Gauss map allows to translate this statement to a purely 2-dimensional problem of finding a balanced geodesic cellulation on the hyperbolic surface, from which the flat metric can be easily recovered. We show next that, given two such flat metrics on the surface, there exists a unique pair of future- and past-complete flat globally hyperbolic maximal ( 2 + 1 ) (2+1) -spacetimes with the same holonomy, in which the flat metrics embed respectively as convex polyhedral Cauchy surfaces. The proof follows from convexity properties of the total length of the associated balanced geodesic cellulations over Teichmüller space.

Source: Journal für die reine und angewandte Mathematik (Crelles Journal)

DOI: 10.1515/crelle-2025-0010

Authors: François Fillastre, Roman Prosanov

Institutions: Centre National de la Recherche Scientifique, Université de Montpellier, Institut Montpelliérain Alexander Grothendieck, University of Vienna

Language: English

Topic: Geometric and Algebraic Topology

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $2,712

Funders: Agence Nationale de la Recherche, Austrian Science Fund