2011/03/13 by Mark Comerford, Comerford, Mark
Engineering · Mathematics · #30C20 #Advanced Numerical Analysis Techniques #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Mathematical Dynamics and Fractals #Primary 30C75 #Secondary 30C45
paper · pdf · doi:10.48550/arxiv.1103.2531
openalex publication_date 2011/03/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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.