The asymptoticity of extremal length in Teichm\"uller space

TL;DR

This paper investigates the asymptotic behavior of extremal length in Teichmüller space, providing explicit limits, formulas for distances between rays, and conditions for asymptoticity, advancing understanding of Teichmüller geometry.

Contribution

It offers explicit formulas for extremal length limits, the limiting Teichmüller distance between rays, and criteria for asymptoticity, extending prior theoretical results.

Findings

Explicit limit formula for extremal length along Teichmüller rays

A new formula for the limiting Teichmüller distance between rays

Necessary and sufficient condition for rays to be asymptotic

Abstract

We study the asymptotic behavior of extremal length along Teichm\"uller rays. Specifically, we determine the limit of extremal length along a Teichm\"uller 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 Teichm\"uller distance between two points moving along arbitrary pairs of Teichm\"uller rays. Furthermore, we derive a necessary and sufficient condition for two Teichm\"uller rays to be asymptotic. Finally, by shifting the initial points of the Teichm\"uller rays along their associated Teichm\"uller geodesics, we show that the minimum of the limiting Teichm\"uller distance coincides with the detour metric between the endpoints of the rays on the horofunction boundary.