Short Separating Geodesics for Multiply Connected Domains

TL;DR

This paper investigates shortest separating hyperbolic geodesics in multiply connected domains, introducing the concept of meridians as shortest simple separating curves, and establishes conditions for their existence and uniqueness.

Contribution

It defines meridians as shortest simple separating geodesics in hyperbolic domains and proves their existence and uniqueness under certain conditions.

Findings

Shortest separating geodesics always exist.

Shortest simple separating geodesics, called meridians, can be found.

Uniqueness of meridians holds when one separating set is connected.

Abstract

We consider the following questions: given a hyperbolic plane domain and a separation of its complement into two disjoint closed sets each of which contains at least two points, what is the shortest closed hyperbolic geodesic which separates these sets and is it a simple closed curve? We show that a shortest curve always exists although in general it may not be simple. However, one can also always find a shortest simple curve and we call such a geodesic a \emph{meridian} of the domain. Meridians generalize to domains of higher connectivity the notion of the equator of an annulus as the shortest geodesic which separates the complement. We show that although they are not in general uniquely defined, if one of the sets of the separation of the complement is connected, then they are unique and are also the shortest possible closed curves which separate the complement in this fashion.