The universal extremal-length constant conjecture for Liouville currents

From papers

Let Σ\Sigma be a hyperbolic surface, let LΣ\mathcal{L}_{\Sigma} be its Liouville current, and let ELΣ\operatorname{EL}_{\Sigma} denote the continuous extension of extremal length to geodesic currents. Universal extremal-length constant conjecture. There is a universal constant CC such that

ELΣ(LΣ)=CArea(Σ).\operatorname{EL}_{\Sigma}(\mathcal{L}_{\Sigma})=C\operatorname{Area}(\Sigma).

This proposes a universal proportionality between the extremal length of the Liouville current and the hyperbolic area of the surface. The supplied text gives no resolution, so the conjecture remains open.

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Sources & referencesView supporting material

Primary source

Dídac Martínez-Granado and Dylan P. Thurston, “From curves to currents”, arXiv:2004.01550 (2024).

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