关于无限连通平面域共形同胚的连续扩张
On continuous extension of conformal homeomorphisms of infinitely connected planar domains
摘要：
考虑广义Jordan域$U$到平面域$Ω$ %（可能是{ \bf无限连通}）上的共形同胚$φ$，满足以下两个条件：（1）$Ω$至多有可数个边界分支是非退化的且它们的直径有有限和;（2）$Ω$的退化边界分量或$U$的退化边界分量构成一个σ有限线性测度集。证明了$φ$连续扩张到$U$的闭包当且仅当$Ω$的每个边界分支局部连通.这推广了Carathéodory连续性定理，并使我们得到了著名的Osgood-Taylor-Carathéodory定理的一个新的推广。有三个问题值得注意。首先，上述条件（1）和（2）都不能被去除。其次，%不需要关于$U$或$Ω$的进一步要求。因此，我们的结果仍然有效的非cofat域，并不遵循从扩展的结果，类似的性质，这是在最近的研究中获得的圆域的共形刚性。最后，当$φ$连续扩张到$U$的闭包时，$Ω$的边界是Peano紧的。因此，我们还证明了以下性质对任何平面域$Ω$都是等价的：（1）$Ω$的边界是Peano紧的. (2)$Ω$具有性质S。(3)$Ω$边界上的每个点都是局部可访问的。(4)$Ω$边界上的每个点都可以本地顺序访问。(5)$Ω$在边界处是串联连接的。(6)$Ω$在Mazurkiewicz距离下的完备化是紧的. \noindent这提供了新的推广早期的部分结果，仅限于特殊情况下，当额外的假设拓扑结构的$U$或其边界是必需的。
英文摘要：
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.
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