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On rigidity and convergence of circle patterns

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Ulrike Bücking

Complex Variables · Submitted 4 May 2016

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Then a subsequence of the corresponding discrete conformal maps converges to a Riemann map between the given domains.

how it is stated · asserted 0.98 read the record →

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arXiv ID1605.01176 (version 2)
Pith ID4OXBXOW3, Pith's permanent record of this paper and its review
CitationUlrike B\"ucking. "On rigidity and convergence of circle patterns." arXiv:1605.01176 (2016). https://pith.science/paper/4OXBXOW3
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The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,887 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.54 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.18 · compares with prior results 0.08 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

Two planar embedded circle patterns with the same combinatorics and the same intersection angles can be considered to define a discrete conformal map. We show that two locally finite circle patterns covering the unit disc are related by a hyperbolic isometry. Furthermore, we prove an analogous rigidity statement for the complex plane if all exterior intersection angles of neighboring circles are uniformly bounded away from $0$. Finally, we study a sequence of two circle patterns with the same combinatorics each of which approximates a given simply connected domain. Assume that all kites are convex and all angles in the kites are uniformly bounded and the radii of one circle pattern converge to $0$. Then a subsequence of the corresponding discrete conformal maps converges to a Riemann map between the given domains.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Two planar embedded circle patterns with the same combinatorics and the same intersection angles can be considered to define a discrete conformal map. background 1.00 0.13 0.06 hedged possibility 0.79
s1 We show that two locally finite circle patterns covering the unit disc are related by a hyperbolic isometry. result 0.98 0.93 0.43 asserted 1.00
s2 Furthermore, we prove an analogous rigidity statement for the complex plane if all exterior intersection angles of neighboring circles are uniformly bounded away from $0$. result 0.96 0.92 0.19 asserted 1.00
s3 Finally, we study a sequence of two circle patterns with the same combinatorics each of which approximates a given simply connected domain. contribution 0.99 0.84 0.04 asserted 0.74
s4 Assume that all kites are convex and all angles in the kites are uniformly bounded and the radii of one circle pattern converge to $0$. method 0.43 0.28 0.03 unclear 0.54
s5 Then a subsequence of the corresponding discrete conformal maps converges to a Riemann map between the given domains. result 1.00 0.87 0.43 asserted 0.98

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v2
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

Two planar embedded circle patterns with the same combinatorics and the same intersection angles can be considered to define a discrete conformal map. We show that two locally finite circle patterns covering the unit disc are related by a hyperbolic isometry. Furthermore, we prove an analogous rigidity statement for the complex plane if all exterior intersection angles of neighboring circles are uniformly bounded away from $0$. Finally, we study a sequence of two circle patterns with the same combinatorics each of which approximates a given simply connected domain. Assume that all kites are convex and all angles in the kites are uniformly bounded and the radii of one circle pattern converge to $0$. Then a subsequence of the corresponding discrete conformal maps converges to a Riemann map between the given domains.

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What this paper does with the arXiv papers it cites

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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