On the extremal length of the hyperbolic metric

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.