Volume 60, pp. 1-19, 2024.
Mobile disks in hyperbolic space and minimization of conformal capacity
Harri Hakula, Mohamed M. S. Nasser, and Matti Vuorinen
Abstract
Our focus is to study constellations of disjoint disks in the hyperbolic
space, i.e., the unit disk equipped with the hyperbolic metric. Each constellation
corresponds to a set which is the union of disks with
hyperbolic radii . The centers of the disks are not
fixed, and hence individual disks of the constellation are allowed to
move under the constraints that they do not overlap and their
hyperbolic radii remain invariant. Our main objective is to find
computational lower bounds for the conformal capacity of a given
constellation. The capacity depends on the centers and radii in a very
complicated way even in the simplest cases when or . In the
absence of analytic methods, our work is based on numerical simulations
using two different numerical methods, the boundary integral equation
method and the -FEM method, respectively. Our simulations combine capacity
computation with minimization methods and produce extremal cases where
the disks of the constellation are grouped next to each other. This
resembles the behavior of animal colonies minimizing
heat flow in arctic areas.
Full Text (PDF) [4.1 MB],
BibTeX
, DOI: 10.1553/etna_vol60s1
Key words
multiply connected domains, hyperbolic geometry, capacity computation
AMS subject classifications
65E05, 31A15, 30C85
Links to the cited ETNA articles
