Uniformization of metric surfaces using isothermal coordinates

Ikonen, Toni

doi:10.54330/afm.112781

Cited by 13 publications

(18 citation statements)

References 30 publications

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“…We refer the reader to Section 3 of [33] for the argument. Analogously, under the reciprocity assumption, we obtain a strengthening of Theorem 1.3 that has been proved by Ikonen [21] for surfaces without boundary.…”

Section: Uniformization Of Metric Surfacessupporting

confidence: 74%

“…The value 4/π is sharp in both Theorems 1.2 and 1.3, as was observed in [41,Example 2.2]. A version of this theorem has been proved by Ikonen [21] for reciprocal surfaces (see Section 1.4) without boundary, using local quasiconformal coordinates in order to construct isothermal coordinates. This approach cannot be employed in our case, since metric surfaces do not have local quasiconformal coordinates, but only weakly quasiconformal parametrizations that are generally not homeomorphic, as provided by Theorem 1.2.…”

Section: Uniformization Of Metric Surfacesmentioning

confidence: 54%

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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: 74%

Section: Uniformization Of Metric Surfacesmentioning

confidence: 54%

Polyhedral approximation and uniformization for non-length surfaces

Ntalampekos¹,

Romney²

2022

Preprint

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“…Rajala's result in particular gives a new approach to the Bonk-Kleiner quasisymmetric uniformization theorem. The results in [3] and [36] have been extended for example in [4], [45], [46], [30], [17]. In [29], Lytchak and the second author provided a further approach to the Bonk-Kleiner theorem which relies on results about the existence and regularity of energy and area minimizing discs in metric spaces admitting a quadratic isoperimetric inequality developed in [26] and [28].…”

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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“…We recall [31,Theorem 1.4] Similarly, Theorem 1.3 of [22] shows that if a metric surface (𝑌, 𝑑 𝑌 ) can be covered by quasiconformal images of domains 𝑉 ⊂ ℝ 2 , then (𝑌, 𝑑 𝑌 ) is quasiconformally equivalent to a Riemannian surface. In particular, we have the following.…”

Section: 4mentioning

confidence: 99%

“…We denote 𝐸 ′ = ι2 (𝜃(𝐸)) ⊂ Z, and apply [22,Theorem 1.3] as in Section 7.1, and find a 1-quasiconformal embedding 𝜓 ∶ Z ⧵ 𝐸 ′ → 𝕊 2 .…”

Section: Points Of Positive Capacitymentioning

confidence: 99%

Two‐dimensional metric spheres from gluing hemispheres

Ikonen

2022

Journal of London Math Soc

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We study metric spheres (𝑍, 𝑑 𝑍 ) obtained by gluing two hemispheres of 𝕊 2 along an orientation-preserving homeomorphism g ∶ 𝕊 1 → 𝕊 1 , where 𝑑 𝑍 is the canonical distance that is locally isometric to 𝕊 2 off the seam. We show that if (𝑍, 𝑑 𝑍 ) is quasiconformally equivalent to 𝕊 2 , in the geometric sense, then g is a welding homeomorphism with conformally removable welding curves.We also show that g is bi-Lipschitz if and only if (𝑍, 𝑑 𝑍 ) has a 1-quasiconformal parametrization whose Jacobian is comparable to the Jacobian of a quasiconformal mapping ℎ ∶ 𝕊 2 → 𝕊 2 . Furthermore, we show that if g −1 is absolutely continuous and g admits a homeomorphic extension with exponentially integrable distortion, then (𝑍, 𝑑 𝑍 ) is quasiconformally equivalent to 𝕊 2 .

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Uniformization of metric surfaces using isothermal coordinates

Cited by 13 publications

References 30 publications

Polyhedral approximation and uniformization for non-length surfaces

Polyhedral approximation and uniformization for non-length surfaces

Quasiconformal almost parametrizations of metric surfaces

Two‐dimensional metric spheres from gluing hemispheres

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