The extremal length systole of the cube punctured at its vertices

Samuel Dobchies; Maxime Fortier Bourque

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arXiv · 2404.00336

The extremal length systole of the cube punctured at its vertices

Samuel Dobchies·Maxime Fortier Bourque

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Abstract

We prove that the extremal length systole of the cube punctured at its vertices is realized by the 12 curves surrounding its edges and give a characterization of the corresponding quadratic differentials, allowing us to estimate its value to high precision. The proof uses a mixture of exact calculations done using branched covers and elliptic integrals, together with estimates obtained using either the geometry of geodesic trajectories on the cube or explicit conformal maps.

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Samuel Dobchies, Maxime Fortier Bourque. 2024-03-30. The extremal length systole of the cube punctured at its vertices. https://arxiv.org/abs/2404.00336

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