Arnold Mathematical Journal

, Volume 3, Issue 1, pp 97–117 | Cite as

Convex Shapes and Harmonic Caps

Research Contribution


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 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.


Julia set Convex shape Polyhedra Harmonic measure Curvature 


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Copyright information

© Institute for Mathematical Sciences (IMS), Stony Brook University, NY 2017

Authors and Affiliations

  1. 1.Department of MathematicsNorthwestern UniversityEvanstonUSA
  2. 2.Department of MathematicsUniversity of ChicagoChicagoUSA

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