Uniformization with Infinitesimally Metric Measures

Rajala, Kai; Rasimus, Martti; Romney, Matthew

doi:10.1007/s12220-021-00689-y

Cited by 6 publications

(6 citation statements)

References 19 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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“…The theorem of Bonk-Kleiner has been extended for example in [20,22,16,17,19]. In the setting of X being an Ahlfors 2-regular and linearly locally connected metric surface, there exist quasisymmetric uniformization results if X\∂X is a domain in S 2 , see [25,18,21], and if X is closed, see [7,10].…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Canonical parametrizations of metric surfaces of higher topology

Fitzi¹,

Meier

2022

Ann. Fenn. Math.

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We give an alternate proof to the following generalization of the uniformization theorem by Bonk and Kleiner. Any linearly locally connected and Ahlfors 2-regular closed metric surface is quasisymmetrically equivalent to a model surface of the same topology. Moreover, we show that this is also true for surfaces as above with non-empty boundary and that the corresponding map can be chosen in a canonical way. Our proof is based on a local argument involving the existence of quasisymmetric parametrizations for metric discs as shown in a paper of Lytchak and Wenger.

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“…Whether such a space exists had been asked by the second-named author as Question 5.6 in [42], and this construction provides an affirmative answer.…”

Section: Uniformization Of Metric Surfacesmentioning

confidence: 97%

“…Various constructions have been studied in detail, including Meyer's snowspheres [34] as well as spheres arising from expanding Thurston maps [5]. See also [42] for an approach to fractal uniformization by adapting conformal modulus methods. 1.4.…”

Section: Uniformization Of Metric Surfacesmentioning

confidence: 99%

Polyhedral approximation and uniformization for non-length surfaces

Ntalampekos¹,

Romney²

2022

Preprint

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We prove that any metric surface (that is, metric space homeomorphic to a 2-manifold with boundary) with locally finite Hausdorff 2-measure is the Gromov-Hausdorff limit of polyhedral surfaces with controlled geometry. We use this result, together with the classical uniformization theorem, to prove that any metric surface homeomorphic to the 2-sphere with finite Hausdorff 2-measure admits a weakly quasiconformal parametrization by the Riemann sphere, answering a question of Rajala-Wenger. These results have been previously established by the authors under the assumption that the metric surface carries a length metric. As a corollary, we obtain new proofs of the uniformization theorems of Bonk-Kleiner for quasispheres and of Rajala for reciprocal surfaces. Another corollary is a simplification of the definition of a reciprocal surface, which answers a question of Rajala concerning minimal hypotheses under which a metric surface is quasiconformally equivalent to a Euclidean domain.

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Uniformization with Infinitesimally Metric Measures

Cited by 6 publications

References 19 publications

Quasiconformal almost parametrizations of metric surfaces

Quasiconformal almost parametrizations of metric surfaces

Canonical parametrizations of metric surfaces of higher topology

Polyhedral approximation and uniformization for non-length surfaces

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