Extensions with bounded ∂-derivative

Reich, Edgar; Chen, Jixiu

doi:10.5186/aasfm.1991.1623

Cited by 23 publications

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“…Reich and Chen [20] proved that F is an extremal quasiconformal deformation if and only if its∂-derivative satisfies the Hamilton-Krushkal condition. The maximal dilatation K[f ] of f can be estimated in terms of the essential supremum of∂F .…”

Section: Extremality Of F (Z T) and F (W T)mentioning

confidence: 99%

Truncations of extremal quasiconformal mappings and their applications

Chen

Fang

2007

Journal of Mathematical Analysis and Applications

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In this paper we give a necessary and sufficient condition to decide whether the Teichmüller equivalency class [α] of a truncation α induced by a uniquely extremal Beltrami differential is a Strebel point in T . We also obtain a necessary and sufficient condition of the unique extremality of α. Using the properties of truncations we provide a method to construct Hamilton sequences. We also get a sufficient condition for the extremality of f (z, t) to be equivalent to that of F (w, t). The corresponding results in the infinitesimal case are obtained, too.

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Section: Extremality Of F (Z T) and F (W T)mentioning

confidence: 99%

Truncations of extremal quasiconformal mappings and their applications

Chen

Fang

2007

Journal of Mathematical Analysis and Applications

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“…For example, the current Lemma 3.1 is conceptually simpler than the corresponding Lemma 4.1 of [I] (and could also have been used to simplify the reasoning used in [I]). The procedure fm estab!ishing the pre!iminary extension lemmas of Section 2. which shall be required later, uses an integral operator previously encountered in [4] and [5].…”

Section: ) S U C~ Thatmentioning

confidence: 99%

A theorem of fehlmann-type for extensions with bounded -derivative

Reich

1995

Complex Variables, Theory and Application: An International Jou

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“…, [5], [7], [8]). (1) f is an extremal quasiconformal mapping iff its complex dilatation µ satisfies the Hamilton-Krushkal condition.…”

Section: Proposition 1 ([4]mentioning

confidence: 99%

“…Similarly, for a continuous function H defined on ∂∆, denote by QD(H) the class of all quasiconformal deformations F on ∆ with boundary values H. GardinerSullivan [3] and Reich-Chen [7] respectively proved that QD(H) is non-empty if and only if H is a Zygmund function, providing that F satisfies the following normalized conditions:…”

Section: §1 Introductionmentioning

confidence: 99%

On the extremality of quasiconformal mappings and quasiconformal deformations

Shen¹

1999

Proc. Amer. Math. Soc.

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Abstract. Given a family of quasiconformal deformations F (w, t) such that ∂F has a uniform bound M , the solution f(z, t)(f (z, 0) = z) of the Löwner-type differential equationis an e 2Mt -quasiconformal mapping. An open question is to determine, for each fixed t > 0, whether the extremality of f (z, t) is equivalent to that of F (w, t). The note gives this a negative approach in both directions.

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Extensions with bounded ∂-derivative

Cited by 23 publications

References 0 publications

Truncations of extremal quasiconformal mappings and their applications

Truncations of extremal quasiconformal mappings and their applications

A theorem of fehlmann-type for extensions with bounded -derivative

On the extremality of quasiconformal mappings and quasiconformal deformations

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