Convex shapes and harmonic caps

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.

We make significant changes to the structure of the article, reordering sections and adjusting definitions. We also added details to clarify arguments