On the number of optimal surfaces

TL;DR

This paper provides exact counts and structural insights into optimal hyperbolic surfaces of genus greater than 3, revealing their automorphism groups and combinatorial properties.

Contribution

It offers explicit formulas for the number of optimal surfaces and characterizes their automorphism groups and combinatorial structures.

Findings

Exact formulas for the number of optimal surfaces of genus > 3

Automorphism groups are cyclic of order 1, 2, 3, or 6

Description of nonorientable hyperbolic optimal surfaces

Abstract

Let 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.