# Conformal Geometry and Dynamics

Published by the American Mathematical Society since 1997, the purpose of this electronic-only journal is to provide a forum for mathematical work in related fields broadly described as conformal geometry and dynamics. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.49.

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by Dimitrios Betsakos, Alexander Solynin and Matti Vuorinen
Conform. Geom. Dyn. 27 (2023), 55-97
DOI: https://doi.org/10.1090/ecgd/381
Published electronically: February 10, 2023

## Abstract:

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 $\mathrm {cap}(E)$ of $E=\cup _{k=0}^n E_k$ behave when the spikes $E_k$, $k=1$, …, $n$, 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 $\mathrm {cap}(E)$ 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, will also be suggested.
References
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Bibliographic Information
• Dimitrios Betsakos
• Affiliation: Department of Mathematics, Aristotle University of Thessaloniki, GR-54124 Thessaloniki, Greece
• MR Author ID: 618946
• Email: betsakos@math.auth.gr
• Alexander Solynin
• Affiliation: Department of Mathematics and Statistics, Texas Tech University, Box 41042, Lubbock, Texas 79409
• MR Author ID: 206458
• Email: alex.solynin@ttu.edu
• Matti Vuorinen
• Affiliation: Department of Mathematics and Statistics, FI-20014 University of Turku, Finland
• MR Author ID: 179630
• ORCID: 0000-0002-1734-8228
• Email: vuorinen@utu.fi
• Received by editor(s): April 30, 2022
• Received by editor(s) in revised form: September 2, 2022
• Published electronically: February 10, 2023

• Dedicated: In memoriam Jukka Sarvas (1944-2021)