Research article 1quasiconformal mappings and cr mappings on. We give a generalization of ahlfors quasiconformal criterion in terms of preschwarzian derivative for sensepreserving harmonic mappings and we use. Ahlforss lectures on quasiconformal mappings ah is the basic literature in the subject. Zajac introiluction in 1 ahlfors investigated the class n of complexvalued. Iwaniec this paper gives an exposition of basic analytical properties of quasiconformal and quasiregular mappings of ndimensional domains. Pdf, notices of the american mathematical society, 53 11. The conformal automorphisms of u form the real mobius group sl2, r, that is, sl2, r modulo its center. Gluing quasiconformal mappings in the plane guowu yao abstract in this paper, several versions of gluing theorems for quasiconformal mappings in the plane are obtained. Jul 14, 2006 lars ahlfors lectures on quasiconformal mappings, based on a course he gave at harvard university in the spring term of 1964, was first published in 1966 and was soon recognized as the classic it was shortly destined to become. Preprint 1 computing quasiconformal maps on riemann surfaces. Quasiconformal mappings quasikonforme abbildung conformal map form mapping ndimensional quasiconformal mappings quasiconformal mapping real analysis. In addition, we examine various properties and computationally explore some examples in r3, enabling a deeper understanding of the mechanics of quasiconformal mappings. He also coined the term quasiconformal inhis1935workonuberlagerungs. Contents introduction topology and analysis conformal mappings in euclidean space the moduli of curve families rings and condensers quasiconformal mappings mapping problems the tukiavaisala extension theorem the mostow rigidity theorem and discrete mobius groups basic notation bibliography index.
Quasiconformal mappings and complex dynamics bgsmath. In this talk, we give a partial answer to this problem by showing that if f has its jacobian j f in muckenhoupt a. On extremal quasiconformal mappings with varying dilatation bounds richard fehlmann and kenichi sakan received september 4, 1985 1. The study folder is kept up to date on the 3rd floor pdf, final version updated on march 3. The measurable riemann mapping theorem and the stoilow factorization. These lectures develop the theory of quasiconformal mappings from scratch, give a selfcontained treatment of the beltrami equation, and cover the basic properties. In this paper we say that a quasiconformal mapping f. On quasiconformal extensions of the beurling ahlfors type clifford j. The inverse of a k quasiconformal homeomorphism is k quasiconformal. The ams bookstore is open, but rapid changes related to the spread of covid 19 may cause delays in delivery services for print products. Geometric construction of quasiconformal mappings in the. Introduction let a homeomorphism of the boundary of the unit disk d z. R s, it follows from the usual compactness properties of quasiconformal mappings that there exists at least one extremal map homotopic to modulo the boundary.
Mathematica volumen 8, 1983, 257324 analytical foundations of the theory of quasiconformal mappings in fin b. This book offers a modern, uptodate introduction to quasiconformal mappings from an explicitly geometric perspective, emphasizing both the extensive developments in mapping theory during the past few decades and the remarkable applications of geometric function theory to other fields, including dynamical systems, kleinian groups, geometric. Mar 27, 2018 the dietrich school of arts and sciences 301 thackeray hall pittsburgh, pa 15260. As an application, we provide a new short proof of the gluing theorem obtained by jiang and qi. Quasiconformal mappings and sharp estimates for the distance. These lectures develop the theory of quasiconformal mappings from scratch, give a selfcontained treatment of the. In this talk, we give a partial answer to this problem by showing that if f has its. Distortion of quasiconformal and quasiregular mappings at. The set of 1 quasiconformal maps forms a group under composition. This book offers a modern, uptodate introduction to quasiconformal mappings from an explicitly geometric perspective, emphasizing both the extensive developments in mapping theory during the past few decades and the remarkable applications of geometric function theory to other fields, including dynamical systems, kleinian groups, geometric topology, differential geometry, and. For a comprehensive treatment, see the book available as ebook from hy university network astalaiwaniecmartin. Relationship between quasisymmetric and quasiconformal maps. Lectures on quasiconformal mappings university lecture.
The book provides an excellent survey of quasiconformal mappings between domains in ndimensional euclidean space, and connections between said quasiconformal mappings and sobolev spaces. On the unique extremality of quasiconformal mappings with dilatation bounds shen, yuliang, tohoku mathematical journal, 2004. Extension theorems for quasiconformal mappings in nspace. An extremal problem for quasiconformal mappings and. Lectures on quasiconformal mappings university lecture series. It grew out as a byproduct and a proper part of our work on the monograph 10 on analytic methods in the. Angles and quasiconformal mappings between manifolds doiserbia. E is a compact subset of s, where kfs\e is the maximal dilatation of fs\e.
On quasiconformal mappings compatible with a fuchsian group kenichi sakan received april 7, 1980 1. Ahlfors used quasiconformal mappings in his geometric approach to nevanlinnas value distribution theory. On the unique extremality of quasiconformal mappings with dilatation bounds shen, yuliang, tohoku mathematical journal, 2004 teichmuller spaces and tame quasiconformal motions jiang, yunping, mitra, sudeb, shiga, hiroshige, and wang, zhe, tohoku mathematical journal, 2018. This process is experimental and the keywords may be updated as the learning algorithm improves.
One of the key tools used in this book is the notion of modulus of families of curves in the euclidean domain. These lectures develop the theory of quasiconformal mappings from scratch, give a selfcontained treatment of the beltrami equation, and cover the basic properties of teichmuller spaces, including the bers embedding and the teichmuller curve. Lectures on quasiconformal mappings lars ahlfors these lectures develop the theory of quasiconformal mappings from scratch, give a selfcontained treatment of the beltrami equation, and cover the basic properties of teichmuller spaces, including the bers embedding and the teichmuller curve. Bibliography includes bibliographical references pages 419425 and index. Conformal and quasiconformal mappings springerlink. Proceedings of the international congress of mathematicians moscow, 1966, 3318. Modulus of a certain curve family given a family f of curves in rn, let admf denote the family of borel mea surable functions. Aug 31, 2006 buy lectures on quasiconformal mappings university lecture series 2rev ed by ahlfors, lars v. Uniform convergence of extremal polynomials when domains have corners and special cusps on the boundary. Mathematica volumen 12, 1987, 185190 remarks on the ahlfors class,t in an anni. The best possibility of gluing quasiconformal mappings is investigated.
Contents introduction topology and analysis conformal mappings in euclidean space the moduli of curve families rings and condensers quasiconformal mappings mapping problems the tukiavaisala extension theorem the mostow rigidity theorem and discrete mobius groups. Strebel 7 has shown that when r is the unit disk such an extremal need not have the form. The theory of quasiconformal mappings in higher dimensions, i. Analytical foundations of the theory of quasiconformal mappings in fin b. Ahlfors, lectures on quasiconformal mappings, van nostrand, princeton, 1996. Let be a univalent sensepreserving harmonic mapping of the open unit disc d z. Preprint 1 computing quasiconformal maps on riemann. We recall that the modulus of f is defined to be f. This program, whose origin can be traced back to the works of m.
The space of k quasiconformal mappings from the complex plane to itself mapping three distinct points to three given points is compact. The following analytic definition for quasiconformal mappings is from 6, theorem. Lectures on ndimensional quasiconformal mappings jussi. These lectures develop the theory of quasiconformal mappings from the beginning on only 146 pages, and give a selfcontained treatment of the beltrami equation, and cover the basic properties of t. Lars ahlforss lectures on quasiconformal mappings, based on a course he gave at harvard university in the spring term of 1964, was first published in 1966 and was soon recognized as the classic it was shortly destined to become. They play a fundamental role in the proof of the mostow.
On quasiconformal harmonic maps pdf luenfai tam and tom y. Quasiconformal harmonic mappings related to janowski. We give a generalization of ahlfors quasiconformal criterion in terms of preschwarzian derivative for sensepreserving harmonic mappings and we use that extend the problem of the univalence of. Quasiconformal mapping singular integral operator riemann mapping beltrami equation distribution derivative these keywords were added by machine and not by the authors. Lars ahlfors lectures on quasiconformal mappings, based on a course he gave at harvard university in the spring term of 1964, was first published in 1966 and. Elliptic pdes and quasiconformal mappings in the plane. Prove or disprove the boundedness of composition operator c f on qspace. Introduction to quasiconformal mappings department of. Introduction in an important paper beurling and ahlfors 6 determined which homeomorphisms of the real line are the boundary values of quasiconformal selfmaps of the upper halfplane.
Lectures on quasiconformal mappings lars ahlfors download. Gluing quasiconformal mappings in the complex plane 3 denote by bels the banach space of beltrami di. In mathematical complex analysis, a quasiconformal mapping, introduced by grotzsch 1928. Ahlfors, lectures on quasiconformal mappings, manuscript prepared with the. Quasiconformal dimensions of selfsimilar fractals tyson, jeremy t. Lars ahlfors lectures on quasiconformal mappings, based on a course he gave at harvard university in the spring term of 1964, was first published in 1966 and was soon recognized as the classic it was shortly destined to become. Abbildung conformal map form mapping ndimensional quasiconformal. Lectures on ndimensional quasiconformal mappings lecture. These lectures develop the theory of quasiconformal mappings from scratch. Analysis of quasiconformal maps in rn usf scholar commons. Note on the distortion theorem for quasiconformal mappings. Quasiconformal mappings and qspaces yuan zhou abstract essenjansonpengxiao 2000 raised the following open problem.
Planar quasiconformal mappings aalto university, department of. A sense preserving homeomorphism f of a domain c is a kquasiconformal. Quasiconformal mappings should carry, unlike derivatives almost everywhere, which often overlook essential features. Mappings with convex potentials and the quasiconformal jacobian problem kovalev, leonid v. On removable sets for quasiconformal mappings 145 2. Pdf quasiconformal maps and teichmuller theoryextremal. He also coined the term quasiconformalinhis1935workonuberlagerungs. An introduction to the theory of higherdimensional. An efficient and novel numerical method for quasiconformal mappings of doubly connected domains daripa p mashat d.
Ahlfors s lectures on quasiconformal mappings, based on a course he gave at harvard university in the spring term of 1964, was first published in 1966 and was soon recognized as the classic it was shortly destined to become. The importance of quasiconformal mappings in complex analysis was realized by ahlfors and teich muller in the 1930s. The lectures in period iii will be based on kari astalas course spring, 20. Everyday low prices and free delivery on eligible orders. Lars ahlfors s lectures on quasiconformal mappings, based on a course he gave at harvard university in the spring term of 1964, was first published in 1966 and was soon recognized as the classic it was shortly destined to become. As the result is local, it also holds true for quasiregular mappings at points at which the mapping is a local homeomorphism.
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