CMC-1 surfaces via osculating M\"{o}bius transformations between circle patterns

TL;DR

This paper introduces a novel method for constructing CMC-1 surfaces in hyperbolic space using osculating Möbius transformations derived from circle patterns, establishing key correspondences and convergence results.

Contribution

It characterizes realizations of dual graphs via osculating Möbius transformations and links them to hyperbolic surface representations, extending the Weierstrass-type representation.

Findings

Established a one-to-one correspondence for circle patterns with shared shear or intersection angles.

Characterized realizations of dual graphs in hyperbolic space.

Proved convergence of the method on triangular lattices.

Abstract

Given two circle patterns of the same combinatorics in the plane, the M\"{o}bius 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 M\"{o}bius 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.