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.
Categories: math.DS math.CV math.MG