## Quasisymmetric parametrizations of two-dimensional metric spheres (2001)

Citations: | 32 - 9 self |

### BibTeX

@MISC{Bonk01quasisymmetricparametrizations,

author = {Mario Bonk and Bruce Kleiner},

title = {Quasisymmetric parametrizations of two-dimensional metric spheres},

year = {2001}

}

### Years of Citing Articles

### OpenURL

### Abstract

### Citations

785 | Algebraic Topology - Spanier - 1966 |

96 |
Quasiconformal maps in metric spaces with controlled geometry
- Heinonen, Koskela
- 1998
(Show Context)
Citation Context ...ur case. Our methods could also easily be adapted to show this directly. From an analytic perspective it is interesting to consider metric spaces that satisfy Poincaré inequalities by assumption (cf. =-=[14, 24, 11, 3, 4, 17]-=-). For an Ahlfors Q-regular 2metric space a (1, Q)-Poincaré inequality is equivalent to the Q-Loewner property as introduced by Heinonen and Koskela [14], see Section 7. It turns out that in dimensio... |

95 |
Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9
- Cheeger
- 1999
(Show Context)
Citation Context ...ur case. Our methods could also easily be adapted to show this directly. From an analytic perspective it is interesting to consider metric spaces that satisfy Poincaré inequalities by assumption (cf. =-=[14, 24, 11, 3, 4, 17]-=-). For an Ahlfors Q-regular 2metric space a (1, Q)-Poincaré inequality is equivalent to the Q-Loewner property as introduced by Heinonen and Koskela [14], see Section 7. It turns out that in dimensio... |

92 | The convergence of circle packings to the Riemann mapping - Rodin, Sullivan - 1987 |

88 |
On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions
- Sullivan
- 1981
(Show Context)
Citation Context ...G admits a discrete, cocompact, and isometric action on hyperbolic 3-space H 3 . This conjecture is a major piece of Thurston’s hyperbolization conjecture for 3-manifolds 2 . By a theorem of Sullivan =-=[28]-=- Cannon’s conjecture is equivalent to the following conjecture: Conjecture 1.3. If G is a hyperbolic group and ∂∞G is homeomorphic to S 2 , then ∂∞G (equipped with one of the metrics mentioned above) ... |

46 |
Ahlfors Q-regular spaces with arbitrary Q > 1 admitting weak Poincaré inequality
- Laakso
(Show Context)
Citation Context ...ur case. Our methods could also easily be adapted to show this directly. From an analytic perspective it is interesting to consider metric spaces that satisfy Poincaré inequalities by assumption (cf. =-=[14, 24, 11, 3, 4, 17]-=-). For an Ahlfors Q-regular 2metric space a (1, Q)-Poincaré inequality is equivalent to the Q-Loewner property as introduced by Heinonen and Koskela [14], see Section 7. It turns out that in dimensio... |

43 |
The uniformization theorem for circle packings
- Beardon, Stephenson
- 1990
(Show Context)
Citation Context ...pace (Z, d) is called λ-linearly locally contractible where λ ≥ 1, if every ball B(a, r) in Z with 0 < r ≤ diam(Z)/λ is contractible inside B(a, λr), i.e., there exists a continuous map H : B(a, r) × =-=[0, 1]-=- → B(a, λr) such that H(·, 0) is the identity on B(a, r) and H(·, 1) is a constant map. The space is called linearly locally contractible, if it is λ-linearly locally contractible for some λ ≥ 1. Simi... |

42 |
Quasisymmetric embeddings of metric spaces
- Tukia, Vaisala
- 1980
(Show Context)
Citation Context ...etric space Z up to quasisymmetry. Particularly interesting are the cases when Z is R n or the standard sphere S n . Quasisymmetric characterizations of R and S 1 have been given by Tukia and Väisälä =-=[31]-=-. If n ≥ 3 then results by Semmes [25] show that natural conditions which one might expect to imply that a metric space is quasisymmetric to S n (or R n ), are in fact insufficient; at present these c... |

33 | Poincaré inequalities and quasiconformal structure on the boundary of some hyperbolic buildings
- Bourdon, Pajot
- 1999
(Show Context)
Citation Context |

31 | Fixed points, Koebe uniformization and circle packings - HE, SCHRAMM - 1993 |

31 | Dimension conforme et sphère à l’infini des variétés à courbure négative - Pansu - 1971 |

30 |
The combinatorial Riemann mapping theorem
- Cannon
- 1994
(Show Context)
Citation Context ...nsmooth setting, in particular in connection with Thurston’s hyperbolization conjecture. This question of nonsmooth uniformization was addressed by Cannon in his combinatorial Riemann mapping theorem =-=[6]-=-. He considers topological surfaces equipped with combinatorial data that lead to a notion of approximate conformal moduli of rings. He then finds conditions on the combinatorial structure that imply ... |

30 | Squaring rectangles: the finite Riemann mapping theorem, The mathematical legacy of Wilhelm Magnus: groups, geometry and special functions - Cannon, Floyd, et al. - 1992 |

29 |
On Thurston’s formulation and proof of Andreev’s theorem. Computational methods and function theory (Valparalo
- Marden, Rodin
- 1989
(Show Context)
Citation Context ... same as the incidence pattern of the graph. Suppose the graph G is the 1-skeleton of a triangulation T of a topological 2sphere. By the Andreev-Koebe-Thurston circle packing theorem (cf. for example =-=[19]-=-) the graph G can be realized as the incidence graph of a circle packing. This means the following. Let V be the vertex set of G with the associated incidence relation ∼. Then there is a family C of p... |

28 |
Plane with A∞-weighted metric not bi-Lipschitz embeddable to
- Laakso
(Show Context)
Citation Context ...ith the metric dα((x1, y1), (x2, y2)) = |x1 − x2| + |y1 − y2| α , where 0 < α < 1, will never be quasisymmetrically homeomorphic to S 2 , see [29, 34]. We also mention that the construction of Laakso =-=[16]-=- provides examples of Ahlfors 2regular, linearly locally contractible 2-spheres which are not bilipschitz homeomorphic to S 2 ; this shows that one cannot replace the word “quasisymmetric” with “bilip... |

23 | Thirty-three yes or no questions about mappings, measures, and metrics - Heinonen, Semmes - 1997 |

21 | Hyperbolic and parabolic packings - He, Schramm - 1995 |

20 |
A quasiconformal group not isomorphic to a Möbius group
- Tukia
- 1981
(Show Context)
Citation Context ...et bilipschitz equivalent to the unit disk B(0, 1) ⊂ R 2 with the metric dα((x1, y1), (x2, y2)) = |x1 − x2| + |y1 − y2| α , where 0 < α < 1, will never be quasisymmetrically homeomorphic to S 2 , see =-=[29, 34]-=-. We also mention that the construction of Laakso [16] provides examples of Ahlfors 2regular, linearly locally contractible 2-spheres which are not bilipschitz homeomorphic to S 2 ; this shows that on... |

20 | Recognizing constant curvature discrete groups in dimension 3 - Cannon, Swenson - 1998 |

19 |
Quasimöbius Maps
- Väisälä
(Show Context)
Citation Context ...this it is easy to see the quasisymmetric invariance of properties like linear local contractibility or linear local connectivity. We list some properties of quasi-Möbius and quasisymmetric maps (cf. =-=[33]-=-): 9(1) Quasi-Möbius and quasisymmetric maps are homeomorphisms onto their image. (2) The composition of an η1-quasi-Möbius map with an η2-quasi-Möbius map is an η2 ◦ η1-quasi-Möbius map. (3) An η-qu... |

18 |
Quasiconformality and quasisymmetry in metric measure spaces
- Tyson
- 1998
(Show Context)
Citation Context ...-sphere as in our theorem satisfies a Poincaré inequality. We will not use this result, since it does not substantially simplify our arguments, and in fact our theorem together with a result by Tyson =-=[32]-=- gives a different way to establish a Poincaré inequality in our case. Our methods could also easily be adapted to show this directly. From an analytic perspective it is interesting to consider metric... |

14 |
Chord-arc surfaces with small constant II
- Semmes
- 1991
(Show Context)
Citation Context ...ht expect to imply that a metric space is quasisymmetric to S n (or R n ), are in fact insufficient; at present these cases look intractable. A result similar to Theorem 1.1 has been proved by Semmes =-=[23]-=- under the additional assumption that Z is a smooth Riemannian surface. The hypothesis of 2regularity in the theorem is essential. A metric 2-sphere containing an open set bilipschitz equivalent to th... |

11 | Hyperbolic Geometry - Cannon, Floyd, et al. - 1997 |

8 | A “regular” pentagonal tiling of the plane - Bowers, Stephenson - 1997 |

5 |
Invariants conformes globaux sur les vari'et'es riemanniennes
- Lelong-Ferrand
- 1973
(Show Context)
Citation Context ...istortion (in fact it is the quasi-Möbius distortion which enters more naturally) via modulus estimates. There are two main ingredients in our implementation of this idea—the Ferrand cross-ratio (cf. =-=[18, 4]-=-), which mediates between the quasisymmetric distortion and the “conformal” distortion, and a modulus comparison proposition which allows one to relate (under suitable conditions) the 2-modulus of a p... |

5 | Quasisymmetric embedding of self similar surfaces and origami with rational maps - Meyer |

3 |
rich families of planar rings
- Sufficiently
- 1999
(Show Context)
Citation Context ... admit a discrete, cocompact, and isometric action on hyperbolic space H 3 . The paper [10] uses [6] and [28, Corollary, p. 468] to give such conditions; the conditions in [10] are in turn applied in =-=[9]-=-. Our Theorems 10.1 or 10.4 can be combined directly with Sullivan’s theorem. The point here is that the action G � ∂∞G of a non-elementary hyperbolic group on its boundary is by uniformly quasi-Möbiu... |

3 | Plane with A1-weighted metric not bilipschitz embeddable to Rn - Laakso |

3 | quasiconformal groups - On - 1986 |

2 |
metric spaces without good parameterizations
- Good
- 1996
(Show Context)
Citation Context ...ticularly interesting are the cases when Z is R n or the standard sphere S n . Quasisymmetric characterizations of R and S 1 have been given by Tukia and Väisälä [31]. If n ≥ 3 then results by Semmes =-=[25]-=- show that natural conditions which one might expect to imply that a metric space is quasisymmetric to S n (or R n ), are in fact insufficient; at present these cases look intractable. A result simila... |

2 | Rigidity for quasi-M"obius group actions - Bonk, Kleiner - 2001 |

2 | lp et espaces de Besov - Cohomologie |

2 | Quasi-M"obius maps - Vaisala - 1988 |

1 |
Rigidity for convergence group actions
- Bonk, Kleiner
- 2000
(Show Context)
Citation Context ...isymmetric to the standard n-sphere. However, if an n-regular n-sphere admits an appropriately large group of symmetries, then it must be quasisymmetrically homeomorphic to the standard n-sphere, see =-=[2]-=-. Theorem 1.1 is closely related to a theorem of Semmes [24] which shows that an Ahlfors n-regular metric space that is a linearly locally contractible topological n-manifold satisfies a (1, 1)-Poinca... |

1 | mappings and strong A1 weights - Bi-Lipschitz - 1993 |

1 | A quasiconformal group not isomorphic to a M"obius group - Tukia - 1981 |