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The infinitesimal and global Thurston geometry of Teichm{ü}ller space

Yi Huang, Ken'Ichi Ohshika, Athanase Papadopoulos

Abstract

We undertake a systematic study of the infinitesimal geometry of the Thurston metric, showing that the topology, convex geometry and metric geometry of the tangent and cotangent spheres based at any marked hyperbolic surface representing a point in Teichm{ü}ller space can recover the marking and geometry of this marked surface. We then translate the results concerning the infinitesimal structures to global geometric statements for the Thurston metric, most notably deriving rigidity statements for the Thurston metric analogous to the celebrated Royden theorem.

The infinitesimal and global Thurston geometry of Teichm{ü}ller space

Abstract

We undertake a systematic study of the infinitesimal geometry of the Thurston metric, showing that the topology, convex geometry and metric geometry of the tangent and cotangent spheres based at any marked hyperbolic surface representing a point in Teichm{ü}ller space can recover the marking and geometry of this marked surface. We then translate the results concerning the infinitesimal structures to global geometric statements for the Thurston metric, most notably deriving rigidity statements for the Thurston metric analogous to the celebrated Royden theorem.
Paper Structure (50 sections, 79 theorems, 159 equations, 15 figures)

This paper contains 50 sections, 79 theorems, 159 equations, 15 figures.

Table of Contents

  1. Introduction
  2. The Thurston metric
  3. Infinitesimal structures on tangent and cotangent space
  4. The unit norm sphere $\mathbf{S}_x$
  5. The unit co-norm sphere $\mathbf{S}_x^*$
  6. Closure correspondence
  7. Topological rigidity
  8. Thurston's stretch maps
  9. Stretch vectors.
  10. Geometric rigidity
  11. Infinitesimal rigidity
  12. Paper outline
  13. Acknowledgements
  14. Preliminaries on Thurston's metric
  15. Lipschitz constants formulation of the Thurston metric
  16. ...and 35 more sections

Key Result

Theorem 1.1

Consider two arbitrary points $x,y\in\mathcal{T}(S)$. The normed vector spaces if and only if the hyperbolic surfaces $(S,x)\text{ and }(S,y)$ are isometric.

Figures (15)

  • Figure 1: A stretch map between two ideal triangles.
  • Figure 2: The four possible configurations of a complete lamination composed of $\gamma_S$, $\alpha_1$ and $\alpha_2$. Case 1 is top left, case 2 is top right, case 3 is bottom left, case 4 is bottom right. We respectively denote the maximal laminations for cases 1 and 3 as $\lambda_-$ and $\lambda_+$. Cases 2 and 4 are those where the two geodesics $\alpha_1$ and $\alpha_2$ spiral around $\gamma_S$ in the same direction, and where we denote the resulting maximal lamination by $\lambda_0$.
  • Figure 3: A lift of $\lambda_0=\gamma_S \cup \alpha_1 \cup \alpha_2$ to the universal cover with specifications of $\tilde{\gamma}_S$, $\tilde{\sigma}$ and $\tilde{\varsigma}$.
  • Figure 4: A lift of $\lambda_+$ to the universal cover with specifications of $\tilde{\gamma}_S$, $\tilde{\sigma}$ and $\tilde{\varsigma}$.
  • Figure 5: The universal cover for an $(4,0)$-crowned annulus. This is a picture when all the $s_i$ are negative.
  • ...and 10 more figures

Theorems & Definitions (187)

  • Theorem 1.1: Infinitesimal rigidity
  • Theorem 1.2: Thurston, ThS
  • Corollary 1.3: contravariant labelling, \ref{['contravariant']}
  • Corollary 1.4: embedded dual curve complex, \ref{['prop:embeddedccx']}
  • Corollary 1.5: embedded curve complex, \ref{['cor:embeddedcx']}
  • Corollary 1.6: covariant labelling, \ref{['inclusion']}
  • Corollary 1.7: \ref{['naturaltransformation']}
  • Theorem 1.8: Topological rigidity
  • Remark 1.9
  • Theorem 1.10: back-time convergence \ref{['thm:backtime']}
  • ...and 177 more