Quasiconformal Mappings in the Plane

ISBN-13: 9783642655159 / Angielski / Miękka / 2011 / 260 str.

Olli Lehto;K.I. Virtanen;K.W. Lucas

Opis

The present text is a fairly direct translation of the German edition "Quasikonforme Abbildungen" published in 1965. During the past decade the theory of quasi conformal mappings in the plane has remained relatively stable. We felt, therefore, that major changes were not necessarily required in the text. In view of the recent progress in the higher-dimensional theory we found it preferable to indicate the two-dimensional case in the title. Our sincere thanks are due to K. W. Lucas, who did the major part of the translation work. In shaping the final form of the text with him we received many valuable suggestions from A. J. Lohwater. We are indebted to Anja Aaltonen and Pentti Dyyster for the prepara- tion of the manuscript, and to Timo Erkama and Tuomas Sorvali for the careful reading and correction of the proofs. Finally, we should like to express our thanks to Springer-Verlag for their friendly coopera- tion in the production of this volume.

Spis treści

I. Geometric Definition of a Quasiconformal Mapping.- to Chapter I.- § 1. Topological Properties of Plane Sets.- § 2. Conformal Mappings of Plane Domains.- § 3. Definition of a Quasiconformal Mapping.- § 4. Conformal Module and Extremal Length.- § 5. Two Basic Properties of Quasiconformal Mappings.- § 6. Module of a Ring Domain.- § 7. Characterization of Quasiconformality with the Help of Ring Domains.- § 8. Extension Theorems for Quasiconformal Mappings.- § 9. Local Characterization of Quasiconformality.- II. Distortion Theorems for Quasiconformal Mappings.- to Chapter II.- §1. Ring Domains with Extremal Module.- § 2. Module of Grötzsch’s Extremal Domain.- § 3. Distortion under a Bounded Quasiconformal Mapping of a Disc.- § 4. Order of Continuity of Quasiconformal Mappings.- § 5. Convergence Theorems for Quasiconformal Mappings.- § 6. Boundary Values of a Quasiconformal Mapping.- § 7. Quasisymmetric Functions.- § 8. Quasiconformal Continuation.- § 9. Circular Dilatation.- III. Auxiliary Results from Real Analysis.- to Chapter III.- § 1. Measure and Integral.- § 2. Absolute Continuity.- § 3. Differentiability of Mappings of Plane Domains.- § 4. Module of a Family of Arcs or Curves.- § 5. Approximation of Measurable Functions.- § 6. Functions with Lp-derivatives.- § 7. Hubert Transformation.- IV. Analytic Characterization of a Quasiconformal Mapping.- to Chapter IV.- § 1. Analytic Properties of a Quasiconformal Mapping.- § 2. Analytic Definition of Quasiconformality.- § 3. Variants of the Geometric Definition.- § 4. Characterization of Quasiconformality with the Help of the Circular Dilatation.- § 5. Complex Dilatationl.- V. Quasiconformal Mappings with Prescribed Complex Dilatation.- to Chapter V.- § l. Existence Theorem.- § 2. Local Dilatation Measures.- § 3. Removable Point Sets.- § 4. Approximation of a Quasiconformal Mapping.- § 5. Application of the Hilbert Transformation to Quasiconformal Mappings21l.- § 6. Conformality at a Point.- § 7. Regularity of a Mapping with Prescribed Complex Dilatation.- VI. Quasiconformal Functions.- to Chapter VI.- § 1. Geometric Characterization of a Quasiconformal Function.- § 2. Analytic Characterization of a Quasiconformal Function.