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Length functions in Teichmüller and anti de Sitter geometry

Filippo Mazzoli, Gabriele Viaggi

TL;DR

This work builds a bridge between Teichmüller theory and 3D anti de Sitter geometry by using Mess representations and pleated surfaces to study length functions. It develops a para-complex shear-bend framework that parametrizes AdS manifolds and yields purely AdS proofs of convexity of length functions along shear paths and second-variation bounds along earthquakes. The approach unifies geodesic-lamination data, boundary dynamics, and intrinsic pleated-surface geometry to derive precise inequalities and variation formulas, extending classical hyperbolic techniques. The results highlight a deep interaction between Teichmüller length data and AdS geometry, with potential implications for dynamics on 3-manifolds and computational tools in Teichmüller theory.

Abstract

We establish a link between the behavior of length functions on Teichmüller space and the geometry of certain anti de Sitter 3-manifolds. As an application, we give new purely anti de Sitter proofs of results of Teichmüller theory such as (strict) convexity of length functions along shear paths and geometric bounds on their second variation along earthquakes. Along the way, we provide shear-bend coordinates for Mess' anti de Sitter 3-manifolds.

Length functions in Teichmüller and anti de Sitter geometry

TL;DR

This work builds a bridge between Teichmüller theory and 3D anti de Sitter geometry by using Mess representations and pleated surfaces to study length functions. It develops a para-complex shear-bend framework that parametrizes AdS manifolds and yields purely AdS proofs of convexity of length functions along shear paths and second-variation bounds along earthquakes. The approach unifies geodesic-lamination data, boundary dynamics, and intrinsic pleated-surface geometry to derive precise inequalities and variation formulas, extending classical hyperbolic techniques. The results highlight a deep interaction between Teichmüller length data and AdS geometry, with potential implications for dynamics on 3-manifolds and computational tools in Teichmüller theory.

Abstract

We establish a link between the behavior of length functions on Teichmüller space and the geometry of certain anti de Sitter 3-manifolds. As an application, we give new purely anti de Sitter proofs of results of Teichmüller theory such as (strict) convexity of length functions along shear paths and geometric bounds on their second variation along earthquakes. Along the way, we provide shear-bend coordinates for Mess' anti de Sitter 3-manifolds.
Paper Structure (47 sections, 34 theorems, 125 equations)

This paper contains 47 sections, 34 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Anti de sitter geometry
  3. Teichmüller and anti de Sitter space
  4. Teichmüller theory
  5. Hyperbolic surfaces
  6. Geodesic laminations
  7. Currents and measured laminations
  8. Length functions
  9. The ${\rm PSL}_2( \mathbb{R} )$ model of $\mathbb{H} ^{2,1}$
  10. Boundary at infinity
  11. Subspaces
  12. Acausal sets and pseudo-metrics
  13. Mess representations and pleated surfaces
  14. Mess representations
  15. Boundary maps
  16. ...and 32 more sections

Key Result

Theorem 1

Let $\rho_{X,Y}$ be a Mess representation. Consider a maximal lamination $\lambda\subset\Sigma$. Let ${\hat{S}_\lambda}\subset \mathcal{CH} _{X,Y}\subset\Omega_{X,Y}$ be the corresponding pleated set. Then: Furthermore, we have for every $\gamma\in\Gamma-\{1\}$, with strict inequality if and only if $\gamma$ intersects the bending locus of $\hat{S}_\lambda$.

Theorems & Definitions (90)

  • Theorem : Theorems A, B, and C of MV22a
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Definition 2.1: Hyperbolic Structures
  • Definition 2.2: Teichmüller Space
  • Definition 2.3: Space of Geodesics
  • Definition 2.4: Geodesic Lamination
  • ...and 80 more