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arXiv · 1602.02327

Convex shapes and harmonic caps

Laura DeMarco·Kathryn Lindsey
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Abstract

Any planar shape $P\subset \mathbb{C}$ can be embedded isometrically as part of the boundary surface $S$ of a convex subset of $\mathbb{R}^3$ such that $\partial P$ supports the positive curvature of $S$. The complement $Q = S \setminus P$ is the associated {\em cap}. We study the cap construction when the curvature is harmonic measure on the boundary of $(\hat{\mathbb{C}}\setminus P, \infty)$. Of particular interest is the case when $P$ is a filled polynomial Julia set and the curvature is proportional to the measure of maximal entropy.

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Laura DeMarco, Kathryn Lindsey. 2016-02-07. Convex shapes and harmonic caps. https://arxiv.org/abs/1602.02327

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