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 induces a $PSL(2,\mathbb{C})$-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 shear coordinates or the same intersection angles. These correspondences are analogous to the Weierstrass representation for surfaces with constant mean curvature $H\equiv 1$ in hyperbolic space. We further establish convergence on triangular lattices.

v3: 33 pages, 9 figures. Final version

References in corpus (4)

• Schwarzian derivatives, projective structures, and the Weil-Petersson gradient flow for renormalized volume
• $C^{\infty }$-convergence of conformal mappings on triangular lattices
• Quadratic differentials and circle patterns on complex projective tori
• Properness for circle packings and Delaunay circle patterns on complex projective structures