Good metric spaces without good parameterizations

Semmes, Stephen

doi:10.4171/rmi/198

Cited by 49 publications

(66 citation statements)

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“…Semmes [31] showed that the theorem of Bonk and Kleiner fails in dimension 3 for a geometrical realization of the decomposition space associated with the Bing double in R 3 , and for the manifold M mentioned earlier. Theorem 1.1 shows that these natural conditions for quasisymmetric parametrization are also insufficient in dimension 4 and higher.…”

Section: Theoremmentioning

confidence: 99%

Quasisymmetric nonparametrization and spaces associated with the Whitehead continuum

Heinonen

2010

Geom. Topol.

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The decomposition space R 3 =Wh associated with the Whitehead continuum Wh is not a manifold, but the product .R 3 =Wh/ R m is homeomorphic to R 3Cm for any m 1 (known since the 1960's). We study the quasisymmetric structure on .R 3 =Wh/ R m and show that the space .R 3 =Wh/ R m may be equipped with a metric resembling R 3Cm geometrically and measure theoretically-it is linearly locally contractible and Ahlfors .3Cm/-regular-nevertheless the resulting space does not admit a quasisymmetric parametrization by R 3Cm . 30C65; 57N16, 57N45In memory of Juha Heinonen, an extraordinary mathematician and friend, who passed away after the mathematical work in this paper was completed. -J-M W

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Section: Theoremmentioning

confidence: 99%

Quasisymmetric nonparametrization and spaces associated with the Whitehead continuum

Heinonen

2010

Geom. Topol.

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“…Work by Semmes [Se1], [Se2] shows that the quasisymmetric characterization of R n or the standard sphere S n for n ≥ 3 is a problem that seems to be beyond reach at the moment. The intermediate case n = 2 is particularly interesting.…”

Section: The Quasisymmetric Uniformization Problemmentioning

confidence: 99%

Quasiconformal geometry of fractals

Bonk¹

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

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Abstract. Many questions in analysis and geometry lead to problems of quasiconformal geometry on non-smooth or fractal spaces. For example, there is a close relation of this subject to the problem of characterizing fundamental groups of hyperbolic 3-orbifolds or to Thurston's characterization of rational functions with finite post-critical set.In recent years, the classical theory of quasiconformal maps between Euclidean spaces has been successfully extended to more general settings and powerful tools have become available. Fractal 2-spheres or Sierpiński carpets are typical spaces for which this deeper understanding of their quasiconformal geometry is particularly relevant and interesting. Mathematics Subject Classification (2000). Primary 30C65; Secondary 20F67.

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“…Examples by Semmes [49] show that the result of Bonk and Kleiner mentioned above does not generalize to dimension 3. Heinonen and Wu [33] and Pankka and Wu [45] gave further examples of geometrically nice spaces without QS parametrizations.…”

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confidence: 99%

Uniformization of two-dimensional metric surfaces

Rajala

2016

Invent. math.

156

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Abstract. We establish uniformization results for metric spaces that are homeomorphic to the Euclidean plane or sphere and have locally finite Hausdorff 2-measure. Applying the geometric definition of quasiconformality, we give a necessary and sufficient condition for such spaces to be QC equivalent to the Euclidean plane, disk, or sphere. Moreover, we show that if such a QC parametrization exists, then the dilatation can be bounded by 2. As an application, we show that the Euclidean upper bound for measures of balls is a sufficient condition for the existence of a 2-QC parametrization. This result gives a new approach to the Bonk-Kleiner theorem on parametrizations of Ahlfors 2-regular spheres by quasisymmetric maps.

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Good metric spaces without good parameterizations

Cited by 49 publications

References 0 publications

Quasisymmetric nonparametrization and spaces associated with the Whitehead continuum

Quasisymmetric nonparametrization and spaces associated with the Whitehead continuum

Quasiconformal geometry of fractals

Uniformization of two-dimensional metric surfaces

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