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The asymptoticity of extremal length in Teichmüller space

Zhiyang Lyu, Yi Qi

TL;DR

The paper advances the understanding of Teichmüller geometry by providing an explicit asymptotic formula for $\mathrm{Ext}$ along Teichmüller rays and, via Kerckhoff's formula, a corresponding limit for the Teichmüller distance between pairs of rays. It clarifies when two rays are asymptotic through a modular equivalence condition on the indecomposable components of the vertical foliations, and connects the minimal limiting distance to the detour metric on the horofunction boundary. By linking extremal length limits, ray asymptotics, and horofunction theory, the work unifies previous Jenkins–Strebel results with general rays and establishes precise criteria for asymptoticity and distance minimization in Teichmüller space. These results have implications for the compactifications of Teichmüller space and the geometry of its boundary, enriching the interplay between extremal length, measured foliations, and horofunction techniques.

Abstract

We study the asymptotic behavior of extremal length along Teichmüller rays. Specifically, we determine the limit of extremal length along a Teichmüller ray and obtain an explicit expression for this limit, which complements a related formula established by Cormac Walsh. Building on this result and Kerckhoff's formula, we establish a formula for the limiting Teichmüller distance between two points moving along arbitrary pairs of Teichmüller rays. Furthermore, we derive a necessary and sufficient condition for two Teichmüller rays to be asymptotic. Finally, by shifting the initial points of the Teichmüller rays along their associated Teichmüller geodesics, we show that the minimum of the limiting Teichmüller distance coincides with the detour metric between the endpoints of the rays on the horofunction boundary.

The asymptoticity of extremal length in Teichmüller space

TL;DR

The paper advances the understanding of Teichmüller geometry by providing an explicit asymptotic formula for along Teichmüller rays and, via Kerckhoff's formula, a corresponding limit for the Teichmüller distance between pairs of rays. It clarifies when two rays are asymptotic through a modular equivalence condition on the indecomposable components of the vertical foliations, and connects the minimal limiting distance to the detour metric on the horofunction boundary. By linking extremal length limits, ray asymptotics, and horofunction theory, the work unifies previous Jenkins–Strebel results with general rays and establishes precise criteria for asymptoticity and distance minimization in Teichmüller space. These results have implications for the compactifications of Teichmüller space and the geometry of its boundary, enriching the interplay between extremal length, measured foliations, and horofunction techniques.

Abstract

We study the asymptotic behavior of extremal length along Teichmüller rays. Specifically, we determine the limit of extremal length along a Teichmüller ray and obtain an explicit expression for this limit, which complements a related formula established by Cormac Walsh. Building on this result and Kerckhoff's formula, we establish a formula for the limiting Teichmüller distance between two points moving along arbitrary pairs of Teichmüller rays. Furthermore, we derive a necessary and sufficient condition for two Teichmüller rays to be asymptotic. Finally, by shifting the initial points of the Teichmüller rays along their associated Teichmüller geodesics, we show that the minimum of the limiting Teichmüller distance coincides with the detour metric between the endpoints of the rays on the horofunction boundary.

Paper Structure

This paper contains 12 sections, 16 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Teichmüller spaces
  4. Measured foliations
  5. Quadratic differentials
  6. Extremal length
  7. Teichmüller rays
  8. The asymptotic behavior of extremal length
  9. The asymptotic behavior of pairs of Teichmüller rays
  10. The minimum of the limiting Teichmüller distance
  11. The horofunction compactification
  12. The detour metric

Key Result

Theorem 1.1

Let $\mathcal{R}_{q,X}(t)$ be the Teichmüller ray from $X\in\mathcal{T}(S)$ determined by the unit-norm quadratic differential $q$. Suppose that the vertical foliation of $q$ can be expressed as $V(q)=\sum_{j=1}^na_jG_j$, where $a_j$ is a positive coefficient and $G_j$ is an indecomposable component where $X_t$ is a point on the Teichmüller ray $\mathcal{R}_{q,X}(t)$.

Theorems & Definitions (27)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Proposition 1.5
  • Corollary 1.6
  • Lemma 2.1: Kerckhoff's formula
  • Remark
  • Lemma 2.2
  • Lemma 3.1: Wal2019
  • ...and 17 more