Spherical and hyperbolic orthogonal ring patterns: integrability and variational principles

TL;DR

This paper introduces orthogonal ring patterns on the sphere and hyperbolic plane, showing their radii follow a discrete integrable system related to Q4, with variational principles and applications to harmonic maps and constant mean curvature surfaces.

Contribution

It generalizes circle patterns to ring patterns on curved surfaces, linking them to integrable systems and providing a variational framework for existence and uniqueness.

Findings

Radii of ring patterns satisfy a discrete integrable system.

Variational principles involve elliptic dilogarithm functions.

Numerical examples and connections to harmonic maps and CMC surfaces.

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.