The Discrete Schwarz-Pick Lemma For Circle Packings Revisited
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917079521689600 |
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| author | Lodha, Arham Rajendra |
| author_facet | Lodha, Arham Rajendra |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_10703 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Discrete Schwarz-Pick Lemma For Circle Packings Revisited Lodha, Arham Rajendra Metric Geometry Geometric Topology 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. |
| title | The Discrete Schwarz-Pick Lemma For Circle Packings Revisited |
| topic | Metric Geometry Geometric Topology |
| url | https://arxiv.org/abs/2511.10703 |