Planar relative Schottky sets and quasisymmetric maps
Planar relative Schottky sets and quasisymmetric maps
复制标题
平面相对肖特基集和拟对称映射
DOI：
10.1112/plms/pdr038
复制
发表时间：
2012
影响因子：
1.8
通讯作者：
S. Merenkov
中科院分区：
文献类型：
--
作者：
S. Merenkov
平面域Ω中的相对肖特基集是Ω的子集，通过从Ω中移除其闭包在Ω中且两两不相交的开几何圆盘而获得。本文研究了零测度的相对Schottky集之间的拟对称映射和相关映射。我们证明，特别是，拟对称映射之间的这些集合在约旦域是共形的，局部双Lipschitz，和他们的一阶导数是局部Lipschitz。我们还提供了Jordan域中相对Schottky集的局部双Lipschitz一致化结果，并建立了单位圆盘中相对Schottky集的局部拟对称映射的刚性。
A relative Schottky set in a planar domain Ω is a subset of Ω obtained by removing from Ω open geometric disks whose closures are in Ω and are pairwise disjoint. In this paper, we study quasisymmetric and related maps between relative Schottky sets of measure zero. We prove, in particular, that quasisymmetric maps between such sets in Jordan domains are conformal, locally bi‐Lipschitz, and that their first derivatives are locally Lipschitz. We also provide a locally bi‐Lipschitz uniformization result for relative Schottky sets in Jordan domains and establish rigidity with respect to local quasisymmetric maps for relative Schottky sets in the unit disk.