On the number of optimal surfaces

Abstract

Let X be a closed oriented Riemann surface of genus 2 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 4 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.