Quasiconformal almost parametrizations of metric surfaces

Damaris, Meier,; Wenger, Stefan

doi:10.48550/arxiv.2106.01256

Cited by 7 publications

(14 citation statements)

References 35 publications

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“…It verifies a conjecture of Rajala and Wenger found as Question 1.3 in [22]. Theorem 1.2 with the additional assumption that the metric on X is a length metric was recently proved by the authors in [37], as well as independently by Meier-Wenger [33].…”

Section: Uniformization Of Metric Surfacessupporting

confidence: 85%

“…It is the case that any metric space X as in Theorem 1.4 is quasiconvex and hence bi-Lipschitz equivalent to a surface with a length metric that is also Ahlfors 2-regular and linearly locally connected; see [44] or [47] for a proof. Thus Theorem 1.4 can also be derived from the weaker version of Theorem 1.2 as given in [33] or [37]. However, our approach allows one to avoid this technical point regarding quasiconvexity.…”

Section: Uniformization Of Metric Surfacesmentioning

confidence: 99%

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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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Section: Uniformization Of Metric Surfacessupporting

confidence: 85%

Section: Uniformization Of Metric Surfacesmentioning

confidence: 99%

Polyhedral approximation and uniformization for non-length surfaces

Ntalampekos¹,

Romney²

2022

Preprint

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“…There are several classes of important non-injective mappings of bounded outer dilatation. For example, one might consider z → z 2 in the complex plane (or other N -to-1 quasiregular mappings, see the monograph [Ric93]), folding maps R n × R : (x, y) → (x, |y|) (or other mappings of bounded deformation, see [Gig18b,GPS20]), or monotone mappings in some metric applications; see [LW17,IR20,MW21,NR21,Iko21a] for some recent work. Many of our results apply to all continuous mappings of bounded outer dilatation.…”

Section: Introductionmentioning

confidence: 99%

Pullback of a quasiconformal map between arbitrary metric measure spaces

Ikonen¹,

Lučić²,

Pasqualetto³

2021

Preprint

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We prove that every (geometrically) quasiconformal homeomorphism between metric measure spaces induces an isomorphism between the cotangent modules constructed by Gigli. We obtain this by first showing that every continuous mapping ϕ with bounded outer dilatation induces a pullback map ϕ * between the cotangent modules of Gigli, and then proving the functorial nature of the resulting pullback operator. Such pullback is consistent with the differential for metric-valued locally Sobolev maps introduced by Gigli-Pasqualetto-Soultanis.Using the consistency between Gigli's and Cheeger's cotangent modules for PI spaces, we prove that quasiconformal homeomorphisms between PI spaces preserve the dimension of Cheeger charts, thereby generalizing earlier work by Heinonen-Koskela-Shanmugalingam-Tyson.Finally, we show that if ϕ is a given homeomorphism with bounded outer dilatation, then ϕ −1 has bounded outer dilatation if and only if ϕ * is invertible and ϕ −1 is Sobolev. In contrast to the setting of Euclidean spaces, Carnot groups, or more generally, Ahlfors regular PI spaces, the Sobolev regularity of ϕ −1 needs to be assumed separately.

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“…This is accomplished in Proposition 3.8. We finally mention that Theorem 1.6 has found another application in the recent article [32], where it was used to prove the existence of a quasiconformal parametrization of 2-dimensional metric surface under minimal conditions. 1.4.…”

mentioning

confidence: 99%

Undistorted fillings in subsets of metric spaces

Basso¹,

Wenger²,

Young³

2021

Preprint

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We prove that if a quasiconvex subset X of a metric space Y has finite Nagata dimension and is Lipschitz k-connected or admits Euclidean isoperimetric inequalities up to dimension k for some k then X is isoperimetrically undistorted in Y up to dimension k + 1. This generalizes and strengthens a recent result of the third named author and has several consequences and applications. It yields for example that in spaces of finite Nagata dimension, Lipschitz connectedness implies Euclidean isoperimetric inequalities, and Euclidean isoperimetric inequalities imply coning inequalities. It furthermore allows us to prove an analog of the Federer-Fleming deformation theorem in spaces of finite Nagata dimension admitting Euclidean isoperimetric inequalities.

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Quasiconformal almost parametrizations of metric surfaces

Cited by 7 publications

References 35 publications

Polyhedral approximation and uniformization for non-length surfaces

Polyhedral approximation and uniformization for non-length surfaces

Pullback of a quasiconformal map between arbitrary metric measure spaces

Undistorted fillings in subsets of metric spaces

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