2024/03/30 by Dobchies, Samuel, Bourque, Maxime Fortier
#Complex Variables (math.CV) #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2404.00336
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.