On removable sets for Sobolev spaces in the plane

Abstract

Let $K$ be a compact subset of $\bar{\mathbf{C}}={\mathbf{R}}^2$ and let $K^c$ denote its complement. We say $K\in HR$, $K$ is holomorphically removable, if whenever $F:\bar{\mathbf{C}}\to \bar{\mathbf{C}}$ is a homeomorphism and $F$ is holomorphic off $K$, then $F$ is a Möbius transformation. By composing with a Möbius transform, we may assume $F(\infty )=\infty$. The contribution of this paper is to show that a large class of sets are $HR$. Our motivation for these results is that these sets occur naturally (e.g. as certain Julia sets) in dynamical systems, and the property of being $HR$ plays an important role in the Douady-Hubbard description of their structure.

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