论文标题
关于无限连接的平面域的保形同态的连续扩展
On continuous extension of conformal homeomorphisms of infinitely connected planar domains
论文作者
论文摘要
我们认为，在平面域上$ω$％（可能是{\ bf无限连接}}的通用Jordan域$ u $的共形同构$φ$，满足了接下来的两个条件：（1）在最数的$ω$的许多边界组件中，$ω$的许多边界组成部分都是非分数的，其直径是有限的。 （2）$ω$的退化边界组件或$ u $的边界组件形成一组Sigma-Finite线性度量。我们证明，当$φ$连续扩展到$ u $的关闭时，并且仅当$ω$的每个边界组件都是本地连接的。这概括了Carathéodory的连续性定理，并使我们对著名的Osgood-Taylor-Carthéodory定理进行了新的概括。有三个值得注意的问题。首先，以上任何条件（1）和（2）均无法删除。其次，％不需要$ u $或$ω$的进一步要求。因此，我们的结果对于非核心域仍然有效，并且不遵循类似性质的扩展结果，这些结果是在最近关于圆形结构域共形刚性的研究中获得的。最后，当$φ$确实不断扩展到$ U $的关闭时，$ω$的边界是Peano Commantum。因此，我们还表明，对于任何平面域$ω$：以下属性是等效的： （1）$ω$的边界是Peano Commantum。 （2）$ω$具有属性S。 （3）$ω$的边界上的每个点都是本地访问的。 （4）$ω$的边界上的每个点都是局部依次访问的。 （5）$ω$在边界有限连接。 （6）在Mazurkiewicz距离下完成$ω$的完成是紧凑的。 \ noindent这提供了限制在特殊情况的早期部分结果的新概括，当需要$ u $或其边界的其他假设。
We consider conformal homeomorphisms $φ$ of generalized Jordan domains $U$ onto planar domains $Ω$ %, possibly {\bf infinitely connected}, 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, %no further requirements concerning $U$ or $Ω$ are needed. So 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 $Ω$: (1) The boundary of $Ω$ is a Peano compactum. (2) $Ω$ has Property S. (3) Every point on the boundary of $Ω$ is locally accessible. (4) Every point on the boundary of $Ω$ is locally sequentially accessible. (5) $Ω$ is finitely connected at the boundary. (6) The completion of $Ω$ under the Mazurkiewicz distance is compact. \noindent This provides new generalizations of earlier partial results that are restricted to special cases, when additional assumptions on the topology of $U$ or its boundary are required.
