  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.
