On the extremal length of the hyperbolic metric

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.

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