Displaying similar documents to “A topological dichotomy with applications to complex analysis”

On a theorem of Lindelöf

Vladimir Gutlyanskii, Olli Martio, Vladimir Ryazanov (2011)

Annales UMCS, Mathematica

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We give a quasiconformal version of the proof for the classical Lindelöf theorem: Let f map the unit disk D conformally onto the inner domain of a Jordan curve C. Then C is smooth if and only if arh f'(z) has a continuous extension to D. Our proof does not use the Poisson integral representation of harmonic functions in the unit disk.

Bounded geometry of quadrilaterals and variation of multipliers for rational maps

Kevin M. Pilgrim (2004)

Fundamenta Mathematicae

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Let Q be the unit square in the plane and h: Q → h(Q) a quasiconformal map. When h is conformal off a certain self-similar set, the modulus of h(Q) is bounded independent of h. We apply this observation to give explicit estimates for the variation of multipliers of repelling fixed points under a "spinning" quasiconformal deformation of a particular cubic polynomial.

Quasiconformal mappings with Sobolev boundary values

Kari Astala, Mario Bonk, Juha Heinonen (2002)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We consider quasiconformal mappings in the upper half space ℝ + n + 1 of ℝ n + 1 , n ≥ 2 , whose almost everywhere defined trace in ℝ n has distributional differential in L n ( ℝ n ) . We give both geometric and analytic characterizations for this possibility, resembling the situation in the classical Hardy space H 1 . More generally, we consider certain positive functions defined on ℝ + n + 1 , called conformal densities. These densities mimic the averaged derivatives of quasiconformal mappings, and we prove analogous trace theorems...

The boundary absolute continuity of quasiconformal mappings (II).

Juha Heinonen (1996)

Revista Matemática Iberoamericana

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In this paper a quite complete picture is given of the absolute continuity on the boundary of a quasiconformal map B → D, where B is the unit 3-ball and D is a Jordan domain in R with boundary 2-rectifiable in the sense of geometric measure theory. Moreover, examples are constructed, for each n ≥ 3, showing that quasiconformal maps from the unit n-ball onto Jordan domains with boundary (n - 1)-rectifiable need not have absolutely continuous boundary values.

Infinite group actions on spheres.

Gaven J. Martin (1988)

Revista Matemática Iberoamericana

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This paper is mainly intended as a survey of the recent work of a number of authors concerning certain infinite group actions on spheres and to raise some as yet unanswered questions. The main thrust of the current research in this area has been to decide what topological and geometrical properties characterise the infinite conformal or Möbius groups. One should then obtain reasonable topological or geometrical restrictions on a subgroup G of the homeomorphism group of a sphere which...