Spherical and hyperbolic orthogonal ring patterns: Integrability and variational principles
Year: 2025
Type: article
Abstract: We introduce orthogonal ring patterns in the 2-sphere and in the hyperbolic plane, consisting of pairs of concentric circles, which generalize circle patterns. We show that their radii are described by a discrete integrable system. This is a special case of the master integrable equation Q4. The variational description is given in terms of elliptic generalizations of the dilogarithm function. They have the same convexity principles as their circle-pattern counterparts. This allows us to prove existence and uniqueness results for the Dirichlet and Neumann boundary value problems. Some examples are computed numerically. In the limit of small smoothly varying rings, one obtains harmonic maps to the sphere and to the hyperbolic plane. A close relation to discrete surfaces with constant mean curvature is explained.
DOI: 10.1090/tran/9544
Author: Alexander Bobenko
Institution: Technische Universität Berlin
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Deutsche Forschungsgemeinschaft
OpenAlex ID: W4415260036