On Codazzi tensors on a hyperbolic surface and flat Lorentzian geometry

TL;DR

This paper explores the relationship between Codazzi tensors on hyperbolic surfaces and flat Lorentzian geometry, providing new proofs and classifications of certain Lorentzian structures and their moduli spaces.

Contribution

It explicitly relates Mess parameters to embedding data via Codazzi tensors and extends the classification of flat Lorentzian structures to surfaces with cone singularities.

Findings

New Lorentzian proof of Goldman's symplectic form result

Classification of flat spacetimes with particles and cone singularities

Generalization of Codazzi tensor decomposition to cone singularities

Abstract

Using global considerations, Mess proved that the moduli space of globally hyperbolic flat Lorentzian structures on $S\times\mathbb{R}$ is the tangent bundle of the Teichm\"uller space of $S$, if $S$ is a closed surface. One of the goals of this paper is to deepen this surprising occurrence and to make explicit the relation between the Mess parameters and the embedding data of any Cauchy surface. This relation is pointed out by using some specific properties of Codazzi tensors on hyperbolic surfaces. As a by-product we get a new Lorentzian proof of Goldman's celebrated result about the coincidence of the Weil-Petersson symplectic form and the Goldman pairing. In the second part of the paper we use this machinery to get a classification of globally hyperbolic flat space-times with particles of angles in $(0,2\pi)$ containing a uniformly convex Cauchy surface. The analogue of Mess' result is achieved showing that the corresponding moduli space is the tangent bundle of the Teichm\"uller space of a punctured surface. To generalize the theory in the case of particles, we deepen the study of Codazzi tensors on hyperbolic surfaces with cone singularities, proving that the well-known decomposition of a Codazzi tensor in a harmonic part and a trivial part can be generalized in the context of hyperbolic metrics with cone singularities.