The Discrete Schwarz-Pick Lemma For Circle Packings Revisited

TL;DR

This paper extends the Discrete Schwarz-Pick Lemma to include circle packings with obtuse intersections and disjoint packings, establishing conditions under which the lemma holds or fails, using a variational approach.

Contribution

It broadens the applicability of the lemma to a wider range of circle packings with inversive distances in (-1,1], introducing new conditions and counterexamples.

Findings

Lemma holds for inversive distances in (-1,1] with additional triangle weight conditions

Lemma fails for disjoint packings with I ≥ 1, demonstrated by a counterexample

Variational principle is key to the proof and extension of the lemma

Abstract

The Discrete Schwarz-Pick Lemma is a discrete analogue of the classical result from complex analysis, arising from the connection between circle packings and conformal maps established by Thurston. Previous works by Beardon-Stephanson and Van Eeuwen proved this lemma for circle packings where circles are tangent or intersect at non-obtuse angles, corresponding to inversive distances $I \in [0,1]$. This paper extends the investigation to circle packings with obtuse intersections ($I \in (-1,0)$) and disjoint packings ($I>1$). We prove that the Discrete Schwarz-Pick Lemma holds for the full range of intersecting circle packings with inversive distances in $(-1,1]$, provided an additional condition on the weights of each triangle is satisfied. The proof relies on a variational principle for circle packings with inversive distances. Conversely, we show that the lemma fails for disjoint circle packings where $I\geq1$. This is demonstrated by constructing a specific counterexample on a triangulated disk with four vertices.