# Mobile disks in hyperbolic space and minimization of conformal capacity

## TL;DR

This paper investigates the conformal capacity of disk arrangements in hyperbolic space, using numerical methods to find lower bounds and extremal configurations, revealing patterns similar to animal colony heat minimization.

## Contribution

It introduces numerical approaches to estimate conformal capacity for complex hyperbolic disk constellations and identifies extremal configurations through capacity minimization.

## Key findings

- Numerical methods effectively estimate capacity bounds.
- Extremal configurations feature disks grouped together.
- Patterns resemble animal colonies minimizing heat flow.

## Abstract

Our focus is to study constellations of disjoint disks in the hyperbolic space, the unit disk equipped with the hyperbolic metric. Each constellation corresponds to a set $E$ which is the union of $m>2$ disks with hyperbolic radii $r_j>0, j=1,...,m$. 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 $m=3$ or $m=4$. 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 $hp$-FEM method, resp. 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.