On the extremal length of the hyperbolic metric

TL;DR

This work proves that the extremal length of the Liouville current on a closed hyperbolic surface is determined by the surface's topology, confirming a conjecture of Martínez-Granado and Thurston. It achieves this by extending length functionals to geodesic currents, employing Hopf ergodic theory, and adapting return-trajectory techniques to relate Ext_X to conformal metrics via a supremum over ell_{\rho}(\cdot)/\sqrt{Area(\rho)}. A central result is Ext_X(L_X) = $\frac{\pi^{2}}{4}$Area$(X)$, equivalently $\frac{\pi^{3}}{2}|\chi(S)|$, and the hyperbolic metric is extremal for L_X but not for any weighted multi-curve. The paper also provides an area-1 diameter bound for extremal metrics and establishes a general upper bound for Ext_X(\mu) for arbitrary currents using a MG–Thurston style return-trajectory construction. These findings strengthen the bridge between Teichmüller theory, geodesic currents, and extremal length, with topology-driven invariants emerging from geometric analysis.

Abstract

For any closed hyperbolic Riemann surface $X$, we show that the extremal length of the Liouville current is determined solely by the topology of \(X\). This confirms a conjecture of Martínez-Granado and Thurston. We also obtain an upper bound, depending only on $X$, for the diameter of extremal metrics on $X$ with area one.

Figures (1)

• Figure 1: The cuffs $\gamma_{i}$'s and seams $\gamma_{\ell,m}^{k}$'s on $X$.

Theorems & Definitions (31)

• Theorem 1.1: Bonahon Bonahon
• Theorem 1.2: c.f. GT
• Remark 1.3
• Theorem 1.4
• Proposition 2.1
• proof
• Definition 2.2
• Proposition 2.3: GT
• Proposition 2.4
• proof