CMC-1 surfaces via osculating Möbius transformations between circle patterns

Abstract

Given two circle patterns of the same combinatorics in the plane, the Möbius transformations mapping circumdisks of one to the other induce a P S L ( 2 , C ) PSL(2,\mathbb {C}) -valued function on the dual graph. Such a function plays the role of an osculating Möbius transformation and induces a realization of the dual graph in hyperbolic space. We characterize the realizations and obtain a one-to-one correspondence in the cases that the two circle patterns share the same discrete conformal structure. These correspondences are analogous to the Weierstrass representation for surfaces with constant mean curvature H ≡ 1 H\equiv 1 in hyperbolic space. We further establish convergence on triangular lattices.