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arxiv:2505.12400
On the extremal length of the hyperbolic metric
Published on May 18, 2025
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Abstract
The extremal length of the Liouville current on a closed hyperbolic Riemann surface is determined by its topology, and an upper bound for the diameter of extremal metrics with area one is obtained.
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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.
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