2025/11/09 by Zhiyang Lyu, Lyu, Zhiyang, Yi Qi +1
Materials Science · Mathematics · #30F60 #32G15 #57M50 #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Quasicrystal Structures and Properties
paper · pdf · doi:10.48550/arxiv.2511.06314
openalex publication_date 2025/11/09 · openalex created_date 2025/11/12 · openalex updated_date 2026/07/28
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.