Quasispheres and metric doubling measures

Lohvansuu, Atte; Rajala, Kai; Rasimus, Martti

doi:10.1090/proc/13971

Cited by 9 publications

(17 citation statements)

References 26 publications

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“…While we work with metric surfaces of locally finite Hausdorff 2-measure in this paper, the uniformization problem has also been studied for fractal spaces, see for example [31], [32], [25], [38], [37]. The aim of our paper is to establish the existence of parametrizations or almost parametrizations with good properties under nearly minimal conditions on X.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Quasiconformal almost parametrizations of metric surfaces

Damaris¹,

Wenger²

2021

Preprint

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We look for minimal conditions on a two-dimensional metric surface X of locally finite Hausdorff 2-measure under which X admits an (almost) parametrization with good geometric and analytic properties. Only assuming that X is locally geodesic, we show that Jordan domains in X of finite boundary length admit a quasiconformal almost parametrization. If X satisfies some further conditions then such an almost parametrization can be upgraded to a geometrically quasiconformal homeomorphism or a quasisymmetric homeomorphism. In particular, we recover Rajala's recent quasiconformal uniformization theorem in the special case that X is locally geodesic as well as Bonk-Kleiner's quasisymmetric uniformization theorem. On the way we establish the existence of Sobolev discs spanning a given Jordan curve in X under nearly minimal assumptions on X and prove the continuity of energy minimizers.

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Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Quasiconformal almost parametrizations of metric surfaces

Damaris¹,

Wenger²

2021

Preprint

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“…In Section 3 we introduce the notion of infinitesimally metric measure on X. These correspond to the metric doubling measures of David and Semmes [5], [10], the correspondence being similar to the one between metrically quasiconformal maps (an infinitesimal condition) and quasisymmetric maps (a global condition). Metric doubling measures can be used to produce quasisymmetric maps via deformation of the metric on X.…”

Section: Introductionmentioning

confidence: 99%

Uniformization of two-dimensional metric surfaces

Rajala

2016

Invent. math.

Self Cite

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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“…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: 70%

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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Quasispheres and metric doubling measures

Abstract: Applying the Bonk-Kleiner characterization of Ahlfors 2-regular quasispheres, we show that a metric two-sphere X is a quasisphere if and only if X is linearly locally connected and carries a weak metric doubling measure, i.e., a measure that deforms the metric on X without much shrinking.

Cited by 9 publications

References 26 publications

Quasiconformal almost parametrizations of metric surfaces

Quasiconformal almost parametrizations of metric surfaces

Uniformization of two-dimensional metric surfaces

Canonical parameterizations of metric disks

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