Rigidity of singular de-Sitter tori with respect to their lightlike bi-foliation

TL;DR

This paper studies the rigidity of de-Sitter tori with singularities, showing they are uniquely determined by their lightlike bi-foliation, linking Lorentzian geometry with topological dynamics.

Contribution

It introduces a notion of constant curvature Lorentzian surfaces with conical singularities and proves a global rigidity result for de-Sitter tori based on their lightlike bi-foliation.

Findings

De-Sitter tori with a single singularity are uniquely determined by their lightlike bi-foliation.

The study connects Lorentzian geometry with topological dynamics.

Rigidity results are analogous to uniformization in Riemannian surfaces.

Abstract

In this paper, we introduce a natural notion of constant curvature Lorentzian surfaces with conical singularities, and provide a large class of examples of such structures. We moreover initiate the study of their global rigidity, by proving that de-Sitter tori with a single singularity of a fixed angle are determined by the topological equivalence class of their lightlike bi-foliation. While this is reminiscent of Troyanov's uniformization results on Riemannian surfaces with conical singularities, the rigidity will come from topological dynamics in the Lorentzian case.