On continuous extension of conformal homeomorphisms of infinitely connected planar domains
On continuous extension of conformal homeomorphisms of infinitely connected planar domains
复制标题
无限连通平面域共形同胚的连续推广
DOI：
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发表时间：
2020-02
期刊：
影响因子：
--
通讯作者：
Xiao-Ting Yao
中科院分区：
文献类型：
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作者：
Jun Luo;Xiao-Ting Yao
考虑平面域Ω上广义Jordan域U的共形同胚φ满足以下两个条件：(1)Ω的最多可数边界分量不简并，且它们的直径有有限和；(2) Ω的简并边界分量或U的简并边界分量构成一组有限线性测度。证明了当且仅当Ω的每个边界分量都是局部连通时，ϕ连续扩展到U的闭包。这推广了carathsamodory的连续性定理，并使我们得到了著名的osgood - taylor - carathsamodory定理的新推广。有三个问题值得注意。首先，上述条件(1)和(2)都不能去除。其次，我们的结果仍然适用于非cofat域，并且不遵循类似性质的扩展结果，这些结果是在最近关于圆域共形刚度的研究中获得的。最后，当φ确实连续扩展到U的闭包时，Ω的边界是一个Peano紧致。因此，我们还证明了以下性质对任何平面域Ω都是等效的：
We consider conformal homeomorphisms ϕ of generalized Jordan domains U onto planar domains Ω that satisfy both of the next two conditions: (1) at most countably many boundary components of Ω are non-degenerate and their diameters have a finite sum; (2) either the degenerate boundary components of Ω or those of U form a set of sigma-finite linear measure. We prove that ϕ continuously extends to the closure of U if and only if every boundary component of Ω is locally connected. This generalizes the Carathéodory’s Continuity Theorem and leads us to a new generalization of the well known Osgood-Taylor-Carathéodory Theorem. There are three issues that are noteworthy. Firstly, none of the above conditions (1) and (2) can be removed. Secondly, our results remain valid for non-cofat domains and do not follow from the extension results, of a similar nature, that are obtained in very recent studies on the conformal rigidity of circle domains. Finally, when ϕ does extend continuously to the closure of U , the boundary of Ω is a Peano compactum. Therefore, we also show that the following properties are equivalent for any planar domain Ω:
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影响因子：
1.4
作者：
C. Carathéodory
通讯作者：
C. Carathéodory
DOI：
10.1090/s0002-9947-1925-1501320-8
发表时间：
1925-04
影响因子：
1.3
作者：
Robert L. Moore
通讯作者：
Robert L. Moore
DOI：
10.1201/b10617-3
发表时间：
2009-11
期刊：
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影响因子：
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作者：
W. Thurston
通讯作者：
W. Thurston
影响因子：
1.1
作者：
Clinton P. Curry
通讯作者：
Clinton P. Curry
影响因子：
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作者：
Shashank V. Joshi
通讯作者：
Shashank V. Joshi