Convex shapes and harmonic caps
Laura DeMarco, Kathryn Lindsey

TL;DR
This paper explores the geometric embedding of planar shapes as convex surface boundaries in three-dimensional space, focusing on harmonic curvature measures, especially for polynomial Julia sets.
Contribution
It introduces a novel approach to constructing convex caps with harmonic curvature, particularly for filled polynomial Julia sets, linking complex dynamics and convex geometry.
Findings
Established a method for isometric embedding of planar shapes with harmonic curvature
Analyzed the case of Julia sets with curvature proportional to maximal entropy measure
Provided insights into the geometric structure of convex caps in complex dynamics
Abstract
Any planar shape can be embedded isometrically as part of the boundary surface of a convex subset of such that supports the positive curvature of . The complement is the associated {\em cap}. We study the cap construction when the curvature is harmonic measure on the boundary of . Of particular interest is the case when is a filled polynomial Julia set and the curvature is proportional to the measure of maximal entropy.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Point processes and geometric inequalities · Analytic and geometric function theory
