圆环域共形映射的刚性理论与连续延拓
Rigidity and continuous extension for conformal maps of circle domains
AI总结：
本文提出了平面区域间共形映射连续或同胚延拓的充分条件，并利用cofat域与CNED集概念，证明了点边界成分为CNED的圆环域具有共形刚性，为He-Schramm刚性猜想提供了最强证据。
摘要：
我们给出了充分条件，使得边界成分为 Jordan 曲线或点的平面区域之间的共形映射能够连续或同胚地延拓到区域的闭包。我们的条件涉及 cofat 域与 CNED 集（即可数极值距离可忽略集）的概念，这些概念由作者近期引入。我们利用此结果来建立一类圆环域的共形刚性。如果每个到另一圆环域的共形映射都是 Möbius 变换的限制，则该圆环域是共形刚性的。我们证明了点边界成分为 CNED 的圆环域是共形刚性的。该结果是所有早期工作中最强的，为 He-Schramm 的刚性猜想提供了实质性证据，将共形刚性与可去性问题联系起来。
英文摘要：
We present sufficient conditions so that a conformal map between planar domains whose boundary components are Jordan curves or points has a continuous or homeomorphic extension to the closures of the domains. Our conditions involve the notions of cofat domains and CNED sets, i.e., countably negligible for extremal distance, recently introduced by the author. We use this result towards establishing conformal rigidity of a class of circle domains. A circle domain is conformally rigid if every conformal map onto another circle domain is the restriction of a Möbius transformation. We show that circle domains whose point boundary components are CNED are conformally rigid. This result is the strongest among all earlier works and provides substantial evidence towards the rigidity conjecture of He-Schramm, relating the problems of conformal rigidity and removability.
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