Orthosystoles and orthokissing numbers

TL;DR

This paper investigates the orthosystole of hyperbolic surfaces with geodesic boundary, characterizing maximizers for one boundary component and constructing surfaces with large orthosystole for multiple boundaries, showing growth rates with genus.

Contribution

It extends Bavard's work by fully characterizing maximizers for a single boundary and constructing large orthosystole surfaces for multiple boundaries, analyzing growth rates.

Findings

Maximizers for a single boundary are fully characterized.

Constructed surfaces with large orthosystole for multiple boundaries.

Orthosystole growth rate matches Bavard's upper bound as genus increases.

Abstract

For hyperbolic surfaces with geodesic boundary, we study the orthosystole, i.e. the length of a shortest essential arc from the boundary to the boundary. We recover and extend work by Bavard completely characterizing the surfaces maximizing the orthosystole in the case of a single boundary component. For multiple boundary components, we construct surfaces with large orthosystole and show that their orthosystole grows, as the genus goes to infinity, at the same rate as Bavard's upper bound.