On the extremal length of the hyperbolic metric

TL;DR

This paper proves that for closed hyperbolic Riemann surfaces, the extremal length of the Liouville current depends only on topology, confirming a conjecture, and provides bounds on extremal metric diameters.

Contribution

It establishes that the extremal length of the Liouville current is topologically determined and offers bounds on extremal metric diameters, confirming a prior conjecture.

Findings

Extremal length of Liouville current depends solely on surface topology.

Confirmed a conjecture by Marte1nez-Granado and Thurston.

Provided an upper bound for the diameter of extremal metrics with area one.

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 Mart\'inez-Granado and Thurston. We also obtain an upper bound, depending only on $X$, for the diameter of extremal metrics on $X$ with area one.