2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/226770Let X be a closed oriented Riemann surface of genus > 1 of constant negative curvature -1. A surface containing a disk of maximal radius is an optimal surface. This paper gives exact formulae for the number of optimal surfaces of genus > 3 up to orientation-preserving isometry. We show that the automorphism group of such a surface is always cyclic of order 1,2,3 or 6. We also describe a combinatorial structure of nonorientable hyperbolic optimal surfaces.This is the version published by Geometry & Topology Monographs on 29 April 2008Geometric TopologyMetric Geometry53C20, 20H10, 53C40text