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Perron's method and spherical ideal circle patterns with prescribed total geodesic curvatures

Lishan Li, Jun Hu, Yi Qi, Yu Sun

Abstract

In this paper, we apply the classical Perron method to give a proof of the existence and uniqueness/rigidity result of a circle pattern on a closed surface equipped with conical spherical metric when prescribed measures of the angles of intersecting circles stay in the range (0,π/2] and total geodesic curvatures are assigned to the circles, which is recently obtained in [3] via Colin de Verdière's variation method. Then we show the convergence of Thurston's algorithm, which adjusts the geodesic curvatures of circles one by one based on the prescribed values for total geodesic curvatures of the circles, to the desired circle pattern in the setting of the result.

Perron's method and spherical ideal circle patterns with prescribed total geodesic curvatures

Abstract

In this paper, we apply the classical Perron method to give a proof of the existence and uniqueness/rigidity result of a circle pattern on a closed surface equipped with conical spherical metric when prescribed measures of the angles of intersecting circles stay in the range (0,π/2] and total geodesic curvatures are assigned to the circles, which is recently obtained in [3] via Colin de Verdière's variation method. Then we show the convergence of Thurston's algorithm, which adjusts the geodesic curvatures of circles one by one based on the prescribed values for total geodesic curvatures of the circles, to the desired circle pattern in the setting of the result.

Paper Structure

This paper contains 7 sections, 9 theorems, 63 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Closed surface with conic spherical metric
  3. Existence and uniqueness of ideal circle pattern on a closed surface equipped with conical spherical metric
  4. Convergence of Thurston algorithm
  5. Preliminary
  6. Proof of Theorem \ref{['Rigidity theorem']}
  7. Convergence of Thurston's algorithm

Key Result

Theorem 1.1

Let $S$ be a closed topological oriented surface and $G$ be a graph on $S$ with the sets of edges and faces denoted by $E$ and $F$, such that each $e \in E$ is weighted by some $\theta_e \in\left(0, \frac{\pi}{2}\right]$ and every $f \in F$ is simply connected. Then $\langle\alpha_f\rangle_{f \in F}

Figures (8)

  • Figure 1: A local picture of a Delaunay circle pattern and its graph.
  • Figure 2: A local picture of an ideal circle pattern and its graph.
  • Figure 3: A conical spherical disk.
  • Figure 4: A conical spherical disk $D_{\alpha}(r)$.
  • Figure 5: Two intersecting conical spherical disks.
  • ...and 3 more figures

Theorems & Definitions (15)

  • Theorem 1.1: Bobenko-Springborn
  • Theorem 1.2: MR4683863
  • Theorem 1.3
  • Lemma 2.1: Spherical cotangent 4-part formula
  • Lemma 2.2: Proposition 2 of MR4683863
  • Lemma 2.3: Monotonicity MR4683863
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 5 more