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Holomorphic Removability of Julia Sets

Jeremy Kahn
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Abstract

Let $f(z) = z^2 + c$ be a quadratic polynomial, with c in the Mandelbrot set. Assume further that both fixed points of f are repelling, and that f is not renormalizable. Then we prove that the Julia set J of f is holomorphically removable in the sense that every homeomorphism of the complex plane to itself that is conformal off of J is in fact conformal on the entire complex plane. As a corollary, we deduce that the Mandelbrot Set is locally connected at such c.

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Jeremy Kahn. 1998-12-31. Holomorphic Removability of Julia Sets. https://arxiv.org/abs/math/9812164

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