标题： 球面和双曲正交环形图案：可积性与变分原理 显示英文标题

标题： Spherical and hyperbolic orthogonal ring patterns: integrability and variational principles

摘要： 我们在2-球面和双曲平面上引入正交环图案，这些图案由同心圆对组成，它们推广了圆图案。 我们证明它们的半径由一个离散可积系统描述。 这是主可积方程Q4的一个特例。 变分描述是通过双曲对数函数的椭圆推广来给出的。 它们具有与相应圆图案相同的凸性原理。 这使我们能够证明狄利克雷和诺伊曼边界值问题的存在性和唯一性结果。 一些例子进行了数值计算。 在小而平滑变化的环的极限下，可以得到映射到球面和双曲平面的调和映射。 解释了与具有常平均曲率的离散曲面之间的密切关系。 显示英文摘要

摘要： 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.