Rigidity of the timelike marked length spectrum and length-twist coordinates of singular de-Sitter tori

Abstract

In this paper, we study the closed timelike geodesics of de-Sitter tori with one singularity and prove their uniqueness in their free homotopy class. We introduce the notion of timelike marked length spectrum of such a torus, and establish its rigidity with respect to the lengths of two homotopy classes of intersection number one. We also construct length-twist coordinates on the deformation space of de-Sitter tori with one singularity.

Figures (5)

• Figure 2.1: Local de-Sitter singularity of angle $\theta$.
• Figure 2.2: A class A singular $\mathbf{dS}^2$-torus $\mathcal{T}_{\theta,x,y}$.
• Figure 4.1: Rectangle $\mathcal{R}_{\theta,k,l}$ of $\mathbf{dS}^2$ with area $\theta$, $\alpha$ lightlike horizontal edges and timelike vertical edges of lengths $(k,l)$.
• Figure 5.1: Lifts of the arcs $\textcolor{DodgerBlue}{\delta}$, $\textcolor{Blue}{\gamma_$\mathsf{b}$}$ and $\gamma_\mathsf{a}^\pm$ in the universal cover $E$ of the annulus $A$ (for $n=3$).
• Figure 5.2: Rectangle $P\subset\mathbf{dS}^2$ bounded by $\textcolor{DodgerBlue}{\delta}$, $\textcolor{Blue}{\gamma_$\mathsf{b}$}$ and $\gamma_\mathsf{a}^\pm$, and its exterior angles at vertices $v_i$.

Theorems & Definitions (31)

• Proposition 1
• Theorem 2
• Theorem 3
• Remark 2.1
• Definition 2.2
• Definition 2.3: mion-moutonRigiditySingularDeSitter2024
• Lemma 2.4: mion-moutonRigiditySingularDeSitter2024
• Definition 2.5: birmanGaussBonnetTheorem2dimensional1984
• Proposition 2.6: birmanGaussBonnetTheorem2dimensional1984
• Proposition 2.7: mion-moutonRigiditySingularDeSitter2024