Mobile disks in hyperbolic space and minimization of conformal capacity

For a fixed integer $m>2$ and $r_j>0, j=1,...,m,$ our focus is to study disjoint disks with hyperbolic radii $r_j$ and the set $E$ which is their union in the hyperbolic space, the unit disk equipped with the hyperbolic metric. Such a set is called a constellation of $m$ disks. 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 this disk 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 in arctic areas minimizing heat flow.