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.
To the memory of Heiner
Zieschang
Keywords
optimal surface, hyperbolic structure,
maximal embedded disk, minimal covering disk
