In this thesis, we study the geometry of planar shapes and their harmonic caps. Specifically, given a compact continuum $P$, we are interested in constructing a planar cap $\hat P$ such that $P$ and $\hat P$ can be glued together along their boundary to form a topological sphere with prescribed curvature distribution. In 2017, DeMarco and Lindsey published the result that such a process is always possible if the curvature distribution is proportional to the harmonic measure associated to the complement of $P$. Alexandrov's uniqueness theorem implies that the topological sphere has a unique realization as the boundary surface of a compact convex subset of $R^3$. Reshetnyak, who is one of Alexandrov's students, provided an alternative perspective of the realization from the complex analytic point of view. We build on these previous works, and characterize harmonic planarity, in which case said convex subset of $R^3$ is entirely contained in a plane, for planar shapes that are Jordan domains or Jordan arcs. We also study computational and numerical methods for Riemann mapping constructions, including the zipper algorithm and the Schwarz-Christoffel transformations. Finally, we implement a numerical cap construction algorithm in Mathematica and in Matlab to generate approximations of harmonic caps. These results help us develop valuable insights into the geometric properties of harmonic caps and the associated convex realizations.

Creator

• Weng, Shuyi

DOI

• https://doi.org/10.21985/n2-6vkx-w466

Subject

• Mathematics

Language

• en

Alternate Identifier

• http://dissertations.umi.com/northwestern:15269
• etdadmin_upload_764499

Keyword

• Alexandrov realization
• harmonic planarity
• harmonic cap
• conformal geometry

Date created

• 2020-01-01

Resource type

• Dissertation

Rights statement

• In Copyright