## Minimal Surfaces from Circle Patterns: Geometry from Combinatorics (2003)

Venue: | Annals of Mathematics |

Citations: | 47 - 10 self |

### BibTeX

@ARTICLE{Bobenko03minimalsurfaces,

author = {Alexander I. Bobenko and Tim Hoffmann and Boris A. Springborn},

title = {Minimal Surfaces from Circle Patterns: Geometry from Combinatorics},

journal = {Annals of Mathematics},

year = {2003},

volume = {164},

pages = {231--264}

}

### OpenURL

### Abstract

The theory of polyhedral surfaces... In this paper, we investigate conformal discretizations of surfaces, i.e. discretizations in terms of circles and spheres, and introduce a new discrete model for minimal surfaces. See Figs. 1 and 2. In comparison with direct methods (see, in particular, [17]), leading usually to triangle meshes, the less intuitive discretizations of the present paper have essential advantages: they respect conformal properties of surfaces, possess a maximum principle, etc...

### Citations

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Citation Context ...ing discs by edges one obtains a combinatorial representation of a circle packing, see Figure 7 (left). In 1936, Koebe published the following remarkable statement about circle packings in the sphere =-=[22]-=-. Theorem 1 (Koebe). For every triangulation of the sphere there is a packing of circles in the sphere such that circles correspond to vertices, and two circles touch if and only if the corresponding ... |

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Citation Context ...ic, then the surface is part of a sphere (or plane) and every conformal parametrization is also a curvature line parametrization. Definition 2 of discrete isothermic surfaces was already suggested in =-=[3]-=-. Roughly speaking, a discrete isothermic surface is a polyhedral surface in 3-space all faces of which are conformal squares. To make this more precise, we use the notion of a “quad-graph” to describ... |

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Citation Context ... surfaces, there are other interesting subclasses of S-isothermic surfaces. In future publications, we plan to treat discrete constant mean curvature surfaces in Euclidean 3-space and Bryant surfaces =-=[7]-=-, [10]. (Bryant surfaces are surfaces with constant mean curvature 1 in hyperbolic 3-space.) Both are special subclasses of isothermic surfaces that can be characterized in terms of surface transforma... |

56 | Variational principles for circle patterns and Koebe’s theorem
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Citation Context ...e minimal S-isothermic surface under consideration, but not only this. This theorem can be made an effective tool in constructing these surfaces. For that purpose, we use a variational principle from =-=[5]-=-, [28] for constructing circle patterns. This principle provides us with a variational description of discrete minimal S-isothermic surfaces and makes possible a solution of some Plateau problems as w... |

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Citation Context ...ure lines under Gauss-map ⇓ cell decomposition of (a branched cover of) the sphere ⇓ orthogonal circle pattern ⇓ Koebe polyhedron ⇓ discrete minimal surface As usual in the theory on minimal surfaces =-=[18]-=-, one starts constructing such a surface with a rough idea of how it should look. To use our method, one should understand its Gauss map and the combinatorics of the curvature line pattern. The image ... |

47 |
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Citation Context ...on the choice of S when the four points lie in a circle. The cross-ratio of four points in R 3 is thus defined up to complex conjugation. (For an equivalent definition involving quaternions, see [3], =-=[15]-=-.) The cross-ratio of four points in R 3 is invariant under Möbius transformations of R 3 . Conversely, if p1, p2, p3, p4 ∈ R 3 have the same cross-ratio (up to complex conjugation) as p ′ 1 , p′ 2 , ... |

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Citation Context ...of the corresponding circle patterns in the sphere can be achieved by elementary methods; see Section 10. In general, the problem is not elementary. Developing methods introduced by Colin de Verdière =-=[9]-=-, the first and third authors have given a constructive proof of the generalized Koebe theorem, which uses a variational principle [5]. It also provides a method for the numerical construction of circ... |

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Citation Context ...ch other orthogonally. This circle pattern is unique up to Möbius transformations. The first published statement and proof of this theorem seems to be contained in [6]. For generalizations, see [25], =-=[24]-=-, and [5], the latter also for a variational proof (see also §8 of this article). Now, mark the centers of the circles with white dots and mark the intersection points, where two touching pairs of cir... |

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Citation Context ...aces are surfaces with constant mean curvature 1 in hyperbolic 3-space.) Both are special subclasses of isothermic surfaces that can be characterized in terms of surface transformations. (See [4] and =-=[16]-=- for a definition of discrete constant mean curvature surfaces in R 3 in terms of transformations of isothermic surfaces. See [17] for the characterization of continuous Bryant surfaces in terms of su... |

24 |
The geometry of finite topology Bryant surfaces
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Citation Context ...aces, there are other interesting subclasses of S-isothermic surfaces. In future publications, we plan to treat discrete constant mean curvature surfaces in Euclidean 3-space and Bryant surfaces [7], =-=[10]-=-. (Bryant surfaces are surfaces with constant mean curvature 1 in hyperbolic 3-space.) Both are special subclasses of isothermic surfaces that can be characterized in terms of surface transformations.... |

23 | Discrete Constant Mean Curvature Surfaces and their
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Citation Context ...of surfaces, i.e. discretizations in terms of circles and spheres, and introduce a new discrete model for minimal surfaces. See Figures 1 and 2. In comparison with direct methods (see, in particular, =-=[23]-=-), leading *Partially supported by the DFG Research Center Matheon “Mathematics for key technologies” and by the DFG Research Unit “Polyhedral Surfaces”. ∗∗ Supported by the DFG Research Center Matheo... |

23 |
The geometry and topology of three-manifolds, Electronic version
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Citation Context ...ethod for the numerical construction of circle patterns (see also [27]). An alternative algorithm was implemented in Stephenson’s program circlepack [12]. It is based on methods developed by Thurston =-=[29]-=-. The first step in both methods is to transfer the problem from the sphere to the plane by a stereographic projection. Then the radii of the circles are calculated. If the radii are known, it is easy... |

21 | Circle packing: Experiments in discrete Analytic function theory
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Citation Context ...s a variational principle [5]. It also provides a method for the numerical construction of circle patterns (see also [27]). An alternative algorithm was implemented in Stephenson’s program circlepack =-=[12]-=-. It is based on methods developed by Thurston [29]. The first step in both methods is to transfer the problem from the sphere to the plane by a stereographic projection. Then the radii of the circles... |

21 | Complete embedded minimal surfaces of finite total curvature, J.Diff.Geom
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Citation Context ...ation. This powerful method allows the construction of important examples. On the other hand, it requires a specific study for each example; and it is difficult to control the embeddedness. Kapouleas =-=[21]-=- proved the existence of new embedded examples using a new method. He considered finitely many catenoids with the same axis and planes orthogonal to this axis and showed that one can desingularize the... |

20 | Rigidity of infinite disk patterns
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- 1999
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Citation Context ...r the circle patterns to use: Take an infinite regular square grid pattern in the plane. It consists of circles with equal radius r and centers on a square grid with spacing √ 2 r. It was shown by He =-=[13]-=- that these patterns are the only embedded and locally finite orthogonal circle patterns with this quad graph. Project it stereographically to the sphere and build the258 A. I. BOBENKO, T. HOFFMANN, ... |

18 | Discretization of surfaces and integrable systems - Bobenko, Pinkall - 1999 |

17 |
Regularity of Minimal Surfaces
- Dierkes, Hildebrandt, et al.
- 2010
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Citation Context ...edron. Dualize to obtain a discrete version of Enneper’s surface. See Figures 1 (left) and 13. 10.2. The higher order Enneper surfaces. As the next example, consider the higher order Enneper surfaces =-=[11]-=-. Their Weierstrass representation has g(z) =z a . One may think of them as Enneper surfaces with an umbilic point in the center. An orthogonal circle pattern analogue of the maps z a was introduced i... |

16 |
How to cage an egg
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Citation Context ...ing each other orthogonally. This circle pattern is unique up to Möbius transformations. The first published statement and proof of this theorem seems to be contained in [6]. For generalizations, see =-=[25]-=-, [24], and [5], the latter also for a variational proof (see also §8 of this article). Now, mark the centers of the circles with white dots and mark the intersection points, where two touching pairs ... |

15 | The C ∞ -convergence of hexagonal disk packings to the Riemann map
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(Show Context)
Citation Context ...ased on Schramm’s approximation result for circle patterns with the combinatorics of the square grid [26]. The best known convergence result for circle patterns is C ∞ -convergence of circle packings =-=[14]-=-. It is an interesting question whether the convergence of discrete minimal surfaces is as good. Because of the convergence, the theory developed in this paper may be used to obtain new results in the... |

15 |
Ueber einige allgemeine Eigenschaften der Minimumsflächen
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Citation Context ...s, it will be completely legitimate to view them as polyhedral surfaces with planar faces because the class of discrete minimal surfaces is not Möbius invariant anyway. The Christoffel transformation =-=[8]-=- (see [15] for a modern treatment) transforms an isothermic surface into a dual isothermic surface. It plays a crucial role in our considerations. For the reader’s convenience, we provide a short proo... |

15 | Möbius geometry of surfaces of constant mean curvature 1 in hyperbolic space
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Citation Context ...an be characterized in terms of surface transformations. (See [4] and [16] for a definition of discrete constant mean curvature surfaces in R 3 in terms of transformations of isothermic surfaces. See =-=[17]-=- for the characterization of continuous Bryant surfaces in terms of surface transformations.) More generally, we believe that the classes of discrete surfaces considered in this paper will be helpful ... |

14 | Teichmüller theory and handle addition for minimal surfaces
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- 2002
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Citation Context ...are known (the Costa-Hoffman-Meeks surface and generalizations [20]), which prove the existence of Kapouleas’ surfaces with given genus, to construct them using conventional methods is very difficult =-=[30]-=-. Our method may be helpful in addressing these problems. At the present time, however, the construction of new minimal surfaces from discrete ones remains a challenge. Apart from discrete minimal sur... |

13 | Variational principles for circle patterns
- Springborn
- 2003
(Show Context)
Citation Context ...imal S-isothermic surface under consideration, but not only this. This theorem can be made an effective tool in constructing these surfaces. For that purpose, we use a variational principle from [5], =-=[28]-=- for constructing circle patterns. This principle provides us with a variational description of discrete minimal S-isothermic surfaces and makes possible a solution of some Plateau problems as well.2... |

11 |
A complete embedded minimal surface in R 3 with genus one and three ends
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Citation Context ... typical problem in the theory of minimal surfaces is to decide whether surfaces with some required geometric properties exist, and to construct them. The discovery of the Costa-Hoffman-Meeks surface =-=[19]-=-, a turning point of the modern theory of minimal surfaces, was based on the Weierstrass representation. This powerful method allows the construction of important examples. On the other hand, it requi... |

9 | Bobenko, Discrete Z γ and Painlevé equations
- Agafonov, I
(Show Context)
Citation Context ... them as Enneper surfaces with an umbilic point in the center. An orthogonal circle pattern analogue of the maps z a was introduced in [4]. Sectors of these circle patterns were proven to be embedded =-=[2]-=-, [1]. Stereographic projection to the sphere followed by dualization leads to S-isothermic analogues of the higher order Enneper surfaces. An S-isothermic higher order Enneper surface with a simple u... |

6 |
Imbedded circle patterns with the combinatorics of the square grid and discrete Painlevé equations, Discrete Comput
- Agafonov
(Show Context)
Citation Context ... as Enneper surfaces with an umbilic point in the center. An orthogonal circle pattern analogue of the maps z a was introduced in [4]. Sectors of these circle patterns were proven to be embedded [2], =-=[1]-=-. Stereographic projection to the sphere followed by dualization leads to S-isothermic analogues of the higher order Enneper surfaces. An S-isothermic higher order Enneper surface with a simple umbili... |

3 | Complete embedded minimal surfaces of …nite total curvature - Ho¤man, Meeks - 1985 |

3 |
Constructing circle patterns using a new functional
- Springborn
- 2003
(Show Context)
Citation Context ...authors have given a constructive proof of the generalized Koebe theorem, which uses a variational principle [5]. It also provides a method for the numerical construction of circle patterns (see also =-=[27]-=-). An alternative algorithm was implemented in Stephenson’s program circlepack [12]. It is based on methods developed by Thurston [29]. The first step in both methods is to transfer the problem from t... |

3 |
minimal surfaces of finite topology
- Embedded
- 1990
(Show Context)
Citation Context ...s existence result is very intuitive, but it gives no lower bound for the genus of the surfaces. Although some examples with lower genus are known (the Costa-Hoffman-Meeks surface and generalizations =-=[20]-=-), which prove the existence of Kapouleas’ surfaces with given genus, to construct them using conventional methods is very difficult [30]. Our method may be helpful in addressing these problems. At th... |

3 |
patterns with the combinatorics of the square grid
- Circle
- 1997
(Show Context)
Citation Context ...he convergence of discrete minimal S-isothermic surfaces to smooth minimal surfaces. The proof is based on Schramm’s approximation result for circle patterns with the combinatorics of the square grid =-=[26]-=-. The best known convergence result for circle patterns is C ∞ -convergence of circle packings [14]. It is an interesting question whether the convergence of discrete minimal surfaces is as good. Beca... |

2 | The geometry of topology Bryant surfaces - Collin, Hauswirth, et al. |

2 | Complete embedded minimal surfaces of total curvature - Kapouleas - 1997 |

1 | Ueber einige allgemeine Eigenschaften der Minimums - Christoel |

1 | Embedded minimal surfaces of topology - Homan, Meeks - 1990 |