### Abstract

The Koebe-Andreev-Thurston theorem states that for any triangulation of a closed orientable surface Σ_{g} of genus g which is covered by a simple graph in the universal cover, there exists a unique metric of curvature 1, 0 or −1 on the surface depending on whether g = 0, 1 or ≥ 2 such that the surface with this metric admits a circle packing with combinatorics given by the triangulation. Furthermore, the circle packing is essentially rigid, that is, unique up to conformal automorphisms of the surface isotopic to the identity. In this paper, we consider projective structures on the surface where circle packings are also defined. We show that the space of projective structures on a surface of genus g ≥ 2 which admits a circle packing contains a neigborhood of the Koebe-Andreev-Thurston structure homeomorphic to ℝ^{6g−6}. We furthemore show that if a circle packing consists of one circle, then the space is globally homeomorphic to ℝ^{6g−6} and that the circle packing is rigid.

Original language | English |
---|---|

Pages (from-to) | 349-397 |

Number of pages | 49 |

Journal | Journal of Differential Geometry |

Volume | 63 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2003 Jan 1 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory
- Geometry and Topology

### Cite this

*Journal of Differential Geometry*,

*63*(3), 349-397. https://doi.org/10.4310/jdg/1090426770

**Circle packings on surfaces with projective structures.** / Kojima, Sadayoshi; Mizushima, Shigeru; Tan, Ser Peow.

Research output: Contribution to journal › Article

*Journal of Differential Geometry*, vol. 63, no. 3, pp. 349-397. https://doi.org/10.4310/jdg/1090426770

}

TY - JOUR

T1 - Circle packings on surfaces with projective structures

AU - Kojima, Sadayoshi

AU - Mizushima, Shigeru

AU - Tan, Ser Peow

PY - 2003/1/1

Y1 - 2003/1/1

N2 - The Koebe-Andreev-Thurston theorem states that for any triangulation of a closed orientable surface Σg of genus g which is covered by a simple graph in the universal cover, there exists a unique metric of curvature 1, 0 or −1 on the surface depending on whether g = 0, 1 or ≥ 2 such that the surface with this metric admits a circle packing with combinatorics given by the triangulation. Furthermore, the circle packing is essentially rigid, that is, unique up to conformal automorphisms of the surface isotopic to the identity. In this paper, we consider projective structures on the surface where circle packings are also defined. We show that the space of projective structures on a surface of genus g ≥ 2 which admits a circle packing contains a neigborhood of the Koebe-Andreev-Thurston structure homeomorphic to ℝ6g−6. We furthemore show that if a circle packing consists of one circle, then the space is globally homeomorphic to ℝ6g−6 and that the circle packing is rigid.

AB - The Koebe-Andreev-Thurston theorem states that for any triangulation of a closed orientable surface Σg of genus g which is covered by a simple graph in the universal cover, there exists a unique metric of curvature 1, 0 or −1 on the surface depending on whether g = 0, 1 or ≥ 2 such that the surface with this metric admits a circle packing with combinatorics given by the triangulation. Furthermore, the circle packing is essentially rigid, that is, unique up to conformal automorphisms of the surface isotopic to the identity. In this paper, we consider projective structures on the surface where circle packings are also defined. We show that the space of projective structures on a surface of genus g ≥ 2 which admits a circle packing contains a neigborhood of the Koebe-Andreev-Thurston structure homeomorphic to ℝ6g−6. We furthemore show that if a circle packing consists of one circle, then the space is globally homeomorphic to ℝ6g−6 and that the circle packing is rigid.

UR - http://www.scopus.com/inward/record.url?scp=0345359898&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0345359898&partnerID=8YFLogxK

U2 - 10.4310/jdg/1090426770

DO - 10.4310/jdg/1090426770

M3 - Article

AN - SCOPUS:0345359898

VL - 63

SP - 349

EP - 397

JO - Journal of Differential Geometry

JF - Journal of Differential Geometry

SN - 0022-040X

IS - 3

ER -