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The reader will, no doubt, wonder why we have not discussed mapping of multiply connected regions. For simply connected regions with more than one boundary point, the Riemann Mapping Theorem shows that \(\mathbb {D}\) provides a canonical domain. For doubly connected regions (e.g., open annuli), there is an infinite one-parameter family of canonical domains. For regions of connectivity n, \(n \geq 3\) , there are \(3 n-6\) parameters according to which two domains of connectivity n must agree in order for them to be mapped into one another.

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A Riemann Mapping Theorem for Two-Connected Domains in the Plane

  • 章节

  • 发表日期：

    2025年

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  • Peter V. Dovbush,
  • Steven G. Krantz

摘要

The reader will, no doubt, wonder why we have not discussed mapping of multiply connected regions. For simply connected regions with more than one boundary point, the Riemann Mapping Theorem shows that \(\mathbb {D}\) provides a canonical domain. For doubly connected regions (e.g., open annuli), there is an infinite one-parameter family of canonical domains. For regions of connectivity n, \(n \geq 3\) , there are \(3 n-6\) parameters according to which two domains of connectivity n must agree in order for them to be mapped into one another.