论文标题
刺猬的保形能力
Conformal capacity of hedgehogs
论文作者
论文摘要
在本文中，我们讨论了有关“刺猬”的共形冷凝器容量的问题，这些问题是紧凑型集合在单位磁盘$ \ mathbb {d} = \ {z：\ {z：\，| z | <1 \} $组成的中央机构$ e_0 $通常是较小的disk的$ e $ e $ $ \ OVILLINE {\ MATHBB {d}} _ r = \ {z：\，| z | | \ le r \ \} $，$ 0 <r <1 $，以及几个spikes $ e_k $，这些spike $ e_k $是radial间隔$ i（α_k）=α_k）= \ {te^te^te^{te^{iα_k} <1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ fe 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 $I(α_k)$ toward the central body if their hyperbolic lengths are preserved during the motion? （2）容量$ {\ rm cap}（e）$如何取决于尖峰$ e_k $之间的角度分布？我们证明了与这些问题有关的几个结果，并讨论了应用对称类型转换以研究刺猬的能力的方法。还将提出几个开放问题，包括关于三维双曲线空间中刺猬的问题的问题。
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 $I(α_k)=\{te^{iα_k}:\,0\le t<1\}$. 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 $I(α_k)$ 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.
