Convex shapes and harmonic caps

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 $P\subset \mathbb{C}$ 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.