2024/09/10 by Alexander I. Bobenko, Bobenko, Alexander I. · 1 citation
Engineering · #52C26 #Advanced Theoretical and Applied Studies in Material Sciences and Geometry #Complex Variables (math.CV) #Differential Geometry (math.DG) #Elasticity and Wave Propagation #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Material Science and Thermodynamics #Mathematical Physics (math-ph) #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.2409.06573
openalex publication_date 2024/09/10 · openalex created_date 2024/10/22 · openalex updated_date 2026/07/28
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.