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Envelopes of the Thurston metric on Teichmüller space

Huiping Pan, Michael Wolf

TL;DR

The paper provides a complete structural and continuity analysis of envelopes in Teichmüller space under Thurston’s asymmetric metric. Using harmonic stretch lines and straightened laminations, it shows Env(X,Y) is a cone over a cone built from chain-recurrent laminations containing Λ(X,Y), and proves Env(X,Y) depends continuously on endpoints. It extends the framework to endpoints on the Thurston boundary, giving a precise shape description for Env(X,η) and a star-shaped accumulation set around η, with a new harmonic-stretch-line criterion: a piecewise harmonic stretch line through (X,λ,γ) is harmonic iff γ∈Cone(λ). The results unify and extend prior punctured-torus work, link envelopes to out/in envelopes, and provide robust, canonical parametrizations via Θ^{out}(λ) and Θ^{in}(λ). Overall, the work advances understanding of Thurston geometry on higher-genus Teichmüller spaces, including boundary behavior and continuity phenomena surrounding envelopes.

Abstract

For the Thurston (asymmetric) metric on Teichmüller space, the defect from being uniquely geodesic is described by the envelope, defined as the union of geodesics from the initial point to the terminal point. Using the harmonic stretch lines we defined recently, we describe the shape of envelopes as a cone over a cone over a space, defined from a topological invariant of the initial and terminal points. In addition, we show that the envelope is always contractible. We prove that envelopes vary continuously with their endpoints. We also provide a parametrization of out-envelopes and in-envelopes in terms of straightened measured laminations complementary to the prescribed maximally stretched laminations. We extend most of these results to the metrically infinite envelopes which have a terminal point on the Thurston boundary, illustrating some of the nuances of these with examples, and describing the accumulation set. Finally, we develop a new characterization of harmonic stretch lines that avoids a limiting process.

Envelopes of the Thurston metric on Teichmüller space

TL;DR

The paper provides a complete structural and continuity analysis of envelopes in Teichmüller space under Thurston’s asymmetric metric. Using harmonic stretch lines and straightened laminations, it shows Env(X,Y) is a cone over a cone built from chain-recurrent laminations containing Λ(X,Y), and proves Env(X,Y) depends continuously on endpoints. It extends the framework to endpoints on the Thurston boundary, giving a precise shape description for Env(X,η) and a star-shaped accumulation set around η, with a new harmonic-stretch-line criterion: a piecewise harmonic stretch line through (X,λ,γ) is harmonic iff γ∈Cone(λ). The results unify and extend prior punctured-torus work, link envelopes to out/in envelopes, and provide robust, canonical parametrizations via Θ^{out}(λ) and Θ^{in}(λ). Overall, the work advances understanding of Thurston geometry on higher-genus Teichmüller spaces, including boundary behavior and continuity phenomena surrounding envelopes.

Abstract

For the Thurston (asymmetric) metric on Teichmüller space, the defect from being uniquely geodesic is described by the envelope, defined as the union of geodesics from the initial point to the terminal point. Using the harmonic stretch lines we defined recently, we describe the shape of envelopes as a cone over a cone over a space, defined from a topological invariant of the initial and terminal points. In addition, we show that the envelope is always contractible. We prove that envelopes vary continuously with their endpoints. We also provide a parametrization of out-envelopes and in-envelopes in terms of straightened measured laminations complementary to the prescribed maximally stretched laminations. We extend most of these results to the metrically infinite envelopes which have a terminal point on the Thurston boundary, illustrating some of the nuances of these with examples, and describing the accumulation set. Finally, we develop a new characterization of harmonic stretch lines that avoids a limiting process.
Paper Structure (40 sections, 70 theorems, 127 equations, 1 figure)

This paper contains 40 sections, 70 theorems, 127 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Overview
  3. Main Results.
  4. Shape and continuity of Envelopes.
  5. A topological description of the ancestor of the envelope.
  6. Envelopes with an endpoint on the Thurston boundary.
  7. A characterization of harmonic stretch lines using the stretch lamination.
  8. Related results
  9. Organization of the paper
  10. Preliminaries
  11. Teichmüller space and the Thurston metric
  12. Neighbourhoods of laminations.
  13. Balls and the Hausdorff topology
  14. Maximally stretched laminations
  15. Envelopes
  16. ...and 25 more sections

Key Result

Theorem 1.1

Suppose $S$ is a closed orientable surface of genus at least two. The envelope $\mathbf{Env}(X,Y)$ varies continuously with $X$ and $Y$ in $\mathcal{T}(S)$.

Figures (1)

  • Figure 1: The straightening $\gamma(c)$ of the geodesic arc $c$.

Theorems & Definitions (154)

  • Theorem 1.1: Continuity
  • Remark 1.2
  • Theorem 1.3: Shapes
  • Theorem 1.4: out-envelopes and in-envelopes
  • Corollary 1.5
  • Remark 1.6
  • Theorem 1.7: Characterization of harmonic stretch lines
  • Theorem 2.1: Thurston1998, Theorem 8.4
  • Remark 2.2
  • Lemma 2.3
  • ...and 144 more