## Combinatorial Ricci flows on surfaces (2003)

Venue: | JOURNAL OF DIFFERENTIAL GEOMETRY |

Citations: | 59 - 10 self |

### BibTeX

@ARTICLE{Chow03combinatorialricci,

author = {Bennett Chow and Feng Luo},

title = {Combinatorial Ricci flows on surfaces},

journal = {JOURNAL OF DIFFERENTIAL GEOMETRY},

year = {2003},

pages = {97--129}

}

### Years of Citing Articles

### OpenURL

### Abstract

### Citations

395 |
Three-manifolds with positive Ricci curvature
- Hamilton
- 1982
(Show Context)
Citation Context ...] on the rigidity of locally finite infinite circle packings from the convexity of the potential function. 6. Is there any 3-dimensional combinatorial analogous of Hamilton’s 3-dimensional Ricci flow =-=[8]-=-?s124 b. chow & f. luo The recent work of David Glickenstein’s UCSD thesis ([6]) seems to indicate that the analog of the Yamabe flow exists in the combinatorial setting in dimension 3 (see also [4]).... |

76 |
The Ricci flow on the 2-sphere
- Chow
- 1991
(Show Context)
Citation Context ... Riemannian metric (X, gij),R. Hamilton in [7] introduced the 2-dimensional Ricci flow defined by the equation dgij dt = −2Kgij where K is the Gaussian curvature of the surface. It is provedin[7] and =-=[3]-=- that for any closed surface with any initial Riemannian metric,the solution of the Ricci flow exists for all time,and after normalizing the solution to have a fixed area,the solution converges to a c... |

71 |
Geometry and topology of 3–manifolds, Princeton University lecture notes
- Thurston
- 1976
(Show Context)
Citation Context ...bolic metric on X so that all vertex angles are 2π.scombinatorial ricci flows 101 Conditions (1.5a) and (1.5b) were obtained by Thurston in his work on circle packing on surfaces. Indeed,according to =-=[14]-=- Theorem 13.7.1, these two conditions are equivalent to the existence of circle packing metric with curvature equal to zero at each vertex. The following result shows that Condition (1.3) is almost th... |

60 |
The Ricci flow on surfaces. Mathematics and general relativity
- Hamilton
- 1986
(Show Context)
Citation Context ...acking theorem is obtained. As another consequence, Ricci flow suggests a new algorithm to find circle packings. 1. Introduction For a compact surface with a Riemannian metric (X, gij),R. Hamilton in =-=[7]-=- introduced the 2-dimensional Ricci flow defined by the equation dgij dt = −2Kgij where K is the Gaussian curvature of the surface. It is provedin[7] and [3] that for any closed surface with any initi... |

56 | Variational principles for circle patterns and Koebe’s theorem
- Bobenko, Springborn
(Show Context)
Citation Context ...,he then produced the convex function and showed that it is proper. In our case,it is the maximum principle and Palais-Smale condition which imply the exponential convergence. Another related work is =-=[1]-=-.s1.6 combinatorial ricci flows 103 The paper is organized as follows. In §2,we collect some known results concerning triangles. Most of the proofs will be deferred to the appendix. In §3,we derive th... |

39 |
A characterization of ideal polyhedra in hyperbolic 3-space
- Rivin
- 1996
(Show Context)
Citation Context ...ive curvature when t is large? 4. Investigate the 2-dimensional combinatorial Ricci flow in the case that the weight is in [0,π). For instance,can one produce a new proof of the results of Igor Rivin =-=[12]-=- using heat equations? 5. We showed that rigidity of the circle packing follows from the strictly convexity of the potential function in this paper (Corollary 3.11). The convexity is based on the main... |

29 |
On Thurston’s formulation and proof of Andreev’s theorem. Computational methods and function theory (Valparalo
- Marden, Rodin
- 1989
(Show Context)
Citation Context ... to a scalar multiplication. 4. Degeneration of circle packing metrics We will recall a related work of [14]. For completeness a slightly different proof is presented. The proof below follows closely =-=[10]-=-. For simplicity,if I = {rj,...rl} is a subset of vertices V ,we will also use I to denote the index set {j,...,l}.scombinatorial ricci flows 115 Proposition 4.1 ([14]). Suppose (T,Φ) is a weighted ge... |

28 |
Private communication
- Cheng
- 2007
(Show Context)
Citation Context ...∆vivjvk by a spherical triangle of edge lengths ri + rj,rj + rk,rk + ri (note that we must impose ri ∈ (0,π) ands102 b. chow & f. luo ri + rj + rk <πin this case). Numerical simulations by L.T. Cheng =-=[2]-=- shows that the situation is very complicated even in the simplest case of triangulations of the 2-sphere,i.e.,the boundary of a tetrahedron. For instance,if we start with the boundary of the regular ... |

20 | Rigidity of infinite disk patterns
- He
- 1999
(Show Context)
Citation Context ...πχ(X)/N . 3.2 The combinatorial Ricci flow We define the combinatorial Ricci flow (with background metric K 2 )to be (3.1) dri(t)/dt = −Kis(ri(t)). The evolution of curvature was first obtained by He =-=[9]-=- for zero weighted triangulations in Euclidean geometry. Proposition 3.2. Under the Ricci flow (3.1), the curvature Ki(t) evolves according to dKi/dt = � (3.2) Cij(Kj − Ki)+λBiKi j∼i where the sum is ... |

9 | Circle packings of maps in polynomial time
- Mohar
- 1997
(Show Context)
Citation Context ...iginal proof of the existence of zero curvature circle packing metric,Thurston gave an algorithm to find the circle packing. This algorithm adjusts the radii at vertices one at a time. It is shown in =-=[11]-=- that Thurston’s algorithm converges in polynomial time to the circle packing. To the best of our knowledge,the existing computer softwares (for instance,[13]) are based on Thurston’s algorithm. It se... |

7 |
Combinatorial scalar curvature and rigidity of ball packings
- Cooper, Rivin
- 1996
(Show Context)
Citation Context ...w [8]?s124 b. chow & f. luo The recent work of David Glickenstein’s UCSD thesis ([6]) seems to indicate that the analog of the Yamabe flow exists in the combinatorial setting in dimension 3 (see also =-=[4]-=-). Appendix. Some computations involving triangles A1. Basic results ontriangles Given a triangle ∆vivjvk in one of the three geometries K 2 = S 2 ,E 2 and H 2 ,let the inner angle at the vi vertex be... |

3 |
Verdiere, Y.: Un principe variationnel pour les empilements de cercles
- De
- 1991
(Show Context)
Citation Context ...ase) and ∂θi/∂rjrj = ∂θj/∂riri (for Euclidean case) where θi is the inner angle at i-th vertex corresponding to ri radius circle (Lemma 2.3). This was first observed in the paper of Colin de Verdiere =-=[5]-=- for the zero weighted triangulation. This makes the Ricci flow variational. Namely, it is essentially a gradient flow of a convex function. We observed that the degeneration condition obtained in Thu... |

1 |
Amaximum principle for combinatorial Yamabe flow
- Glickenstein
(Show Context)
Citation Context ... of the potential function. 6. Is there any 3-dimensional combinatorial analogous of Hamilton’s 3-dimensional Ricci flow [8]?s124 b. chow & f. luo The recent work of David Glickenstein’s UCSD thesis (=-=[6]-=-) seems to indicate that the analog of the Yamabe flow exists in the combinatorial setting in dimension 3 (see also [4]). Appendix. Some computations involving triangles A1. Basic results ontriangles ... |

1 |
Circle packing software
- Stephenson
(Show Context)
Citation Context ...at vertices one at a time. It is shown in [11] that Thurston’s algorithm converges in polynomial time to the circle packing. To the best of our knowledge,the existing computer softwares (for instance,=-=[13]-=-) are based on Thurston’s algorithm. It seems Ricci flow is more natural and may produce a faster algorithm to find circle packing metrics. 1.4 In the case that the background metric is spherical,we c... |

1 |
private communications
- Li-Tien
(Show Context)
Citation Context ...realize each triangle ∆vivjvk by a spherical triangle of edge lengths ri + rj, rj + rk, rk+ri (note that we must impose ri ∈ (0, π) and ri+rj+rk < π in this case). Numerical simulations by L.T. Cheng =-=[Che]-=- shows that the situation is very complicated even in the simplest case of triangulations of the 2-sphere, i.e., the boundary of a tetrahedron. For instance, if we start with the boundary of the regul... |

1 |
de Verdière, Yves: Un principe variationnel pour les empilements de cercles
- Colin
- 1991
(Show Context)
Citation Context ...ase) and ∂θi/∂rjrj = ∂θj/∂riri (for Euclidean case) where θi is the inner angle at i-th vertex corresponding to ri radius circle (lemma 2.3). This was first observed in the paper of Colin de Verdiere =-=[CV]-=- for the zero weighted triangulation. This makes the Ricci flow variational. Namely, it is essentially a gradient flow of a convex function. 4We observed that the degeneration condition obtained in T... |

1 | A maximum principle in evolutions by combinatorial scalar curvature - Glickenstein - 2002 |