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Hyperbolic Circle Packings and Total Geodesic Curvatures on Surfaces with Boundary

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Guangming Hu, Yi Qi, Yu Sun, Puchun Zhou

Geometric Topology · Submitted 17 Nov 2023

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

The authors' main claim By analyzing limit behaviours of generalized circle packings on polygons, we give an existence and rigidity for the generalized hyperbolic circle packing with conical singularities regarding the total geodesic curvature on each vertex of the contact graph.

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arXiv ID2311.10528 (version 1)
Pith IDQ7QMCUCK, Pith's permanent record of this paper and its review
CitationGuangming Hu, Puchun Zhou, Yi Qi, Yu Sun. "Hyperbolic Circle Packings and Total Geodesic Curvatures on Surfaces with Boundary." arXiv:2311.10528 (2023). https://pith.science/paper/Q7QMCUCK
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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, 4,452 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.08 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.74 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

This paper investigates a generalized hyperbolic circle packing (including circles, horocycles or hypercycles) with respect to the total geodesic curvatures on the surface with boundary. We mainly focus on the existence and rigidity of circle packing whose contact graph is the $1$-skeleton of a finite polygonal cellular decomposition, which is analogous to the construction of Bobenko and Springborn [4]. Motivated by Colin de Verdi\`ere's method [6], we introduce the variational principle for generalized hyperbolic circle packings on polygons. By analyzing limit behaviours of generalized circle packings on polygons, we give an existence and rigidity for the generalized hyperbolic circle packing with conical singularities regarding the total geodesic curvature on each vertex of the contact graph. As a consequence, we introduce the combinatoral Ricci flow to find a desired circle packing with a prescribed total geodesic curvature on each vertex of the contact graph.

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 This paper investigates a generalized hyperbolic circle packing (including circles, horocycles or hypercycles) with respect to the total geodesic curvatures on the surface with boundary. contribution 0.90 0.68 0.06 asserted 0.92
s1 We mainly focus on the existence and rigidity of circle packing whose contact graph is the $1$-skeleton of a finite polygonal cellular decomposition, which is analogous to the construction of Bobenko and Springborn [4]. contribution 0.91 0.61 0.09 asserted 0.88
s2 Motivated by Colin de Verdi\`ere's method [6], we introduce the variational principle for generalized hyperbolic circle packings on polygons. method 0.81 0.91 0.04 asserted 1.00
s3 By analyzing limit behaviours of generalized circle packings on polygons, we give an existence and rigidity for the generalized hyperbolic circle packing with conical singularities regarding the total geodesic curvature on each vertex of the contact graph. result 0.90 0.94 0.48 asserted 1.00
s4 As a consequence, we introduce the combinatoral Ricci flow to find a desired circle packing with a prescribed total geodesic curvature on each vertex of the contact graph. contribution 0.55 0.93 0.32 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.85 · evidence from proof or derivation 1.00.

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  • T body sentences
  • C citations
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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

This paper investigates a generalized hyperbolic circle packing (including circles, horocycles or hypercycles) with respect to the total geodesic curvatures on the surface with boundary. We mainly focus on the existence and rigidity of circle packing whose contact graph is the $1$-skeleton of a finite polygonal cellular decomposition, which is analogous to the construction of Bobenko and Springborn [4]. Motivated by Colin de Verdi\`ere's method [6], we introduce the variational principle for generalized hyperbolic circle packings on polygons. By analyzing limit behaviours of generalized circle packings on polygons, we give an existence and rigidity for the generalized hyperbolic circle packing with conical singularities regarding the total geodesic curvature on each vertex of the contact graph. As a consequence, we introduce the combinatoral Ricci flow to find a desired circle packing with a prescribed total geodesic curvature on each vertex of the contact graph.

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2 arXiv papers cite this in 8 sentences

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

Papers citing it · 6 sentences without a confident label

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What it cites · 8 sentences without a confident label

Cites as background 2 papers
  • The Ricci flow was introduced by Hamilton [ref] and Perelman improved Hamilton’s program of Ricci flow to solve the Poincaré conjecture and Thurston’s geometrization conjecture [ref] .

    math.DG/0303109 · Background · 0.96

  • The Ricci flow was introduced by Hamilton [ref] and Perelman improved Hamilton’s program of Ricci flow to solve the Poincaré conjecture and Thurston’s geometrization conjecture [ref] .

    math.DG/0307245 · Background · 0.96

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