Conformal capacity of hedgehogs

TL;DR

This paper investigates how the conformal capacity of hedgehog-shaped sets in the unit disk varies with the movement and arrangement of their spikes, using symmetrization methods and exploring related open problems.

Contribution

It provides new insights into the behavior of conformal capacity of hedgehogs as spikes move and are arranged, introducing symmetrization techniques and discussing extensions to hyperbolic space.

Findings

Capacity decreases as spikes move toward the center with preserved hyperbolic length.

Capacity depends on the angular distribution of the spikes.

Methods of symmetrization are effective in studying hedgehog capacities.

Abstract

In this paper we discuss problems concerning the conformal condenser capacity of "hedgehogs", which are compact sets $E$ in the unit disk $\mathbb{D}=\{z:\,|z|<1\}$ consisting of a central body $E_0$ that is typically a smaller disk $\overline{\mathbb{D}}_r=\{z:\,|z|\le r\}$, $0<r<1$, and several spikes $E_k$ that are compact sets lying on radial intervals $I(\alpha_k)=\{te^{i\alpha_k}:\,0\le t<1\}$. The main questions we are concerned with are the following: (1) How does the conformal capacity ${\rm cap}(E)$ of $E=\cup_{k=0}^n E_k$ behave when the spikes $E_k$, $k=1,\ldots,n$, move along the intervals $I(\alpha_k)$ toward the central body if their hyperbolic lengths are preserved during the motion? (2) How does the capacity ${\rm cap}(E)$ depend on the distribution of angles between the spikes $E_k$? We prove several results related to these questions and discuss methods of applying symmetrization type transformations to study the capacity of hedgehogs. Several open problems, including problems on the capacity of hedgehogs in the three-dimensional hyperbolic space, also will be suggested.