The asymptoticity of extremal length in Teichmüller space

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.

💡 Research Summary

The paper investigates the long‑time behavior of extremal length along Teichmüller rays and uses this information to obtain precise formulas for the limiting Teichmüller distance between two points moving on arbitrary pairs of rays. Building on Walsh’s earlier work, the authors first extend the known limit formula for extremal length. For a Teichmüller ray (R_{q,X}(t)) determined by a unit‑norm quadratic differential (q) with vertical foliation (V(q)=\sum_{j=1}^{n} a_j G_j) (where each (G_j) is an indecomposable component), they prove that for any measured foliation (F)

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