CMC-1 surfaces via osculating Möbius transformations between circle patterns

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.