Quasisymmetric Koebe uniformization

Merenkov, Sergei; Wildrick, Kevin

doi:10.4171/rmi/743

Cited by 31 publications

(35 citation statements)

References 22 publications

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“…This result has been extended in several consequent works cf. [12], [40], [59], [60]. There the Ahlfors regularity condition is combined with varying geometric conditions on the space Y .…”

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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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“…This result has been extended in several consequent works cf. [12], [40], [59], [60]. There the Ahlfors regularity condition is combined with varying geometric conditions on the space Y .…”

mentioning

confidence: 99%

Uniformization of two-dimensional metric surfaces

Rajala

2016

Invent. math.

156

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“…We refer to Section 2 for the definitions of quasisymmetric homeomorphism, linear local connectedness, and Ahlfors 2-regularity. This result has since been extended, for example, in [5], [34], [47], [48], and [26], and recently in the important paper [37]. We refer to [37] for details and more references.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 69%

Canonical parameterizations of metric disks

Lytchak¹,

Wenger²

2020

Duke Math. J.

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We use the recently established existence and regularity of area and energy minimizing disks in metric spaces to obtain canonical parameterizations of metric surfaces. Our approach yields a new and conceptually simple proof of a well-known theorem of Bonk and Kleiner on the existence of quasisymmetric parameterizations of linearly locally connected, Ahlfors 2-regular metric 2-spheres. Generalizations and applications to the geometry of such surfaces are described.

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“…Other parametrization results for metric surfaces have been obtained for example in [23,36,39,40,43]. These results however do not include a uniqueness part.…”

Section: Resultsmentioning

confidence: 99%

Maximal metric surfaces and the Sobolev-to-Lipschitz property

Creutz

Soultanis

2020

Calc. Var.

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We find maximal representatives within equivalence classes of metric spheres. For Ahlfors regular spheres these are uniquely characterized by satisfying the seemingly unrelated notions of Sobolev-to-Lipschitz property, or volume rigidity. We also apply our construction to solutions of the Plateau problem in metric spaces and obtain a variant of the associated intrinsic disc studied by Lytchak–Wenger, which satisfies a related maximality condition.

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Quasisymmetric Koebe uniformization

Cited by 31 publications

References 22 publications

Uniformization of two-dimensional metric surfaces

Uniformization of two-dimensional metric surfaces

Canonical parameterizations of metric disks

Maximal metric surfaces and the Sobolev-to-Lipschitz property

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