Cross-ratio distortion and Douady-Earle extension: I. A new upper bound on quasiconformality

Hu, Jun; Muzician, Oleg

doi:10.1112/jlms/jds013

Cited by 9 publications

(9 citation statements)

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“…Such result is the analogue for minimal Lagrangian extensions of [HM12, Theorem 5], which proves that the Douady-Earle extension of ϕ λ stays exponentially far from being extremal as λ → +∞. More precisely, Hu and Muzician showed (see [HM12,Theorem 3]) that, if f DE ϕ λ denotes the Douady-Earle extension of ϕ λ , then…”

Section: Discussionmentioning

confidence: 81%

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On the maximal dilatation of quasiconformal minimal Lagrangian extensions

Seppi

2019

Geom Dedicata

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Given a quasisymmetric homeomorphism ϕ of the circle, Bonsante and Schlenker proved the existence and uniqueness of the minimal Lagrangian extension fϕ : H 2 → H 2 to the hyperbolic plane. By previous work of the author, its maximal dilatation satisfies log K(fϕ) ≤ C||ϕ||cr, where ||ϕ||cr denotes the cross-ratio norm. We give constraints on the value of an optimal such constant C, and discuss possible lower inequalities, by studying two one-parameter families of minimal Lagrangian extensions in terms of maximal dilatation and cross-ratio norm.

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Section: Discussionmentioning

confidence: 81%

“…log K(f BA ϕ ) ≤ ||ϕ|| cr + log 2 . Concerning the Douady-Earle extension, in [HM12] Hu and Muzician proved the inequality:…”

Section: Introductionmentioning

confidence: 97%

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On the maximal dilatation of quasiconformal minimal Lagrangian extensions

Seppi

2019

Geom Dedicata

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“…Furthermore, Φ(f ) is quasiconformal if f admits a quasiconformal extension ( [3]). It is proved in [5] that the maximal dilatation K(Φ(f )) on the unit disk D depends on ||f || cr in a linear fashion. Using the expressions ( 9)-( 16), Markovic developed a criterion (Lemma 3.6, [9]) for a family of circle homeomorphisms f to have a uniform upper bound for the maximal dilatations of their Douady-Earle extensions on a uniform neighborhood of the origin.…”

Section: Now We Can Easily See |Cmentioning

confidence: 99%

Cross-ratio distortion and Douady–Earle extension: III. How to control the dilatation near the origin

Hu¹,

Muzician²

2019

Ann. Acad. Sci. Fenn. Math.

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In this paper, we study how the maximal dilatation of the Douady-Earle extension near the origin is controlled by the distortion of the boundary map on finitely many points. Consider the case of points evenly spread on the circle. We show that the maximal dilatation of the extension in a neighborhood of the origin has an upper bound only depending on the cross-ratio distortion of the boundary map on these points if and only if the number n of the points is more than 4. Furthermore, we show that the size of the neighborhood is universal for each n ≥ 5 in the sense that its size only depends on the distortion.This property implies that for each n ≥ 5, {Φ(f )| U } f ∈Fn is a normal family under certain normalization.To prove these two results, it suffices to show the following theorem.

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“…For some recent applications of these extensions, we refer to [5], [17], [18], [21] and [6]. Further investigation on regularities or generalizations of the Douady-Earle extensions to large classes of circle maps have been developed in [11]- [15] and [16].…”

Section: Introductionmentioning

confidence: 99%

Characterization of the asymptotic Teichmüller space of the open unit disk through shears

Fan

2014

Pure and Applied Mathematics Quarterly

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We give a parametrization to the asymptotic Teichmüller space AT (D) of the open unit disk D through equivalent classes of shear functions induced by quasisymmetric homeomorphisms on the Farey tesselation of D. Then using the parametrization, we define a new metric on AT (D). Two other related metrics are also introduced on AT (D) by using cross-ratio distortions or quadrilateral dilatations under the boundary maps on degenerating sequences of quadruples or quadrilaterals. We show that the topologies induced by the three metrics are equivalent to the one induced by the Teichmüller metric on AT (D). Before proving our main results, we revisit and rectify a mistake in the proof in [21] on the characterization of quasisymmetric homeomorphisms in terms of shear functions.2000 Mathematics Subject Classification. 30C75, 30F60.

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Cross-ratio distortion and Douady-Earle extension: I. A new upper bound on quasiconformality

Cited by 9 publications

References 0 publications

On the maximal dilatation of quasiconformal minimal Lagrangian extensions

On the maximal dilatation of quasiconformal minimal Lagrangian extensions

Cross-ratio distortion and Douady–Earle extension: III. How to control the dilatation near the origin

Characterization of the asymptotic Teichmüller space of the open unit disk through shears

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