Rigidity and continuous extension for conformal maps of circle domains
Rigidity and continuous extension for conformal maps of circle domains
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DOI：
10.1090/tran/8923
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发表时间：
2022-10
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影响因子：
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通讯作者：
Dimitrios Ntalampekos
中科院分区：
文献类型：
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作者：
Dimitrios Ntalampekos
我们提出了足够的条件，使得边界分量为乔丹曲线或点的平面域之间的共形映射具有到域闭包的连续或同胚扩展。我们的条件涉及 cofat 域和 CNED 集的概念，即作者最近引入的极值距离可忽略不计。我们利用这一结果来建立一类圆域的共形刚性。如果到另一个圆域的每个共形映射都是 M\"obius 变换的限制，则圆域是共形刚性的。我们证明点边界分量为 CNED 的圆域是共形刚性的。这个结果是所有早期工作中最强的，并且为 He-Schramm 的刚性猜想提供了实质性证据，涉及共形刚性和可移除性问题。
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 M\"obius 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.