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On the number of optimal surfaces

Alina Vdovina

Geometry & Topology Monographs 14 (2008) 557–567

DOI: 10.2140/gtm.2008.14.557

arXiv: 0904.1877

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.

To the memory of Heiner Zieschang

Keywords

optimal surface, hyperbolic structure, maximal embedded disk, minimal covering disk

Mathematical Subject Classification

Primary: 53C20

Secondary: 20H10, 53C40

References
Publication

Received: 31 July 2006
Revised: 28 July 2007
Accepted: 28 July 2007
Published: 29 April 2008

Authors
Alina Vdovina
School of Mathematics and Statistics
Newcastle University
Newcastle-upon-Tyne NE1 7RU
UK