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

Martin Mion-Mouton

TL;DR

The work introduces and analyzes singular Lorentzian surfaces of constant curvature, focusing on de-Sitter tori with a single conical singularity. It develops a robust framework linking geometry to 1D dynamics via lightlike foliations, HIET-based polygon constructions, and a surgery toolkit to navigate the deformation space. A principal achievement is a rigidity result: de-Sitter tori with one singularity are determined by the topological class of their lightlike foliations, with deformation data captured by projective asymptotic cycles. The paper also builds a structured deformation space, proves its Hausdorff property, and provides existence results for families realizing prescribed asymptotic-cycle data, paving the way for a Lorentzian analogue of uniformization in this singular setting.

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.

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

TL;DR

The work introduces and analyzes singular Lorentzian surfaces of constant curvature, focusing on de-Sitter tori with a single conical singularity. It develops a robust framework linking geometry to 1D dynamics via lightlike foliations, HIET-based polygon constructions, and a surgery toolkit to navigate the deformation space. A principal achievement is a rigidity result: de-Sitter tori with one singularity are determined by the topological class of their lightlike foliations, with deformation data captured by projective asymptotic cycles. The paper also builds a structured deformation space, proves its Hausdorff property, and provides existence results for families realizing prescribed asymptotic-cycle data, paving the way for a Lorentzian analogue of uniformization in this singular setting.

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.
Paper Structure (68 sections, 80 theorems, 139 equations, 5 figures)

This paper contains 68 sections, 80 theorems, 139 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Singular de-Sitter surfaces
  3. Geometric rigidity of lightlike bifoliations
  4. Global description of the deformation space in terms of asymptotic cycles
  5. Methods, and strategies of the main proofs
  6. Perspectives on multiple singularities
  7. Connection with the smoothness of conjugacies for circle diffeomorphisms with breaks
  8. Organization of the paper
  9. Constant curvature Lorentzian surfaces
  10. Lorentzian surfaces, time and space-orientation, and lightlike foliations
  11. The Minkowski space
  12. The de-Sitter space
  13. Lorentzian $(\mathbf{G},\mathbf{X})$-surfaces
  14. Cyclic order, intervals of a circle and rectangles of $\mathbf{dS}^2$
  15. Singular constant curvature Lorentzian surfaces
  16. ...and 53 more sections

Key Result

Theorem 1

Let $S_1$ and $S_2$ be two singular $\mathbf{dS}^2$-tori having a unique singularity of the same angle and minimal lightlike foliations. Then any topological equivalence between the lightlike bi-foliations of $S_1$ and $S_2$ is an isometry.

Figures (5)

  • Figure 3.1: Standard singularity, quadrants and orientations conventions.
  • Figure 4.1: $\mathbf{dS}^2$-torus with one singularity and a closed $\alpha$-lightlike leaf.
  • Figure 4.2: $\mathbf{dS}^2$-torus with one singularity and two minimal foliations.
  • Figure A.1: Maximizing timelike curves avoid positive singularities.
  • Figure A.2: Shadow for maximizing timelike curves at a negative singularity.

Theorems & Definitions (182)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Definition 2.4
  • ...and 172 more