Conformally Invariant Metrics and Quasiconformal Mappings

1. Front Matter

   Pages i-xix

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2. Introduction and Review

   1. Front Matter

      Pages 1-1

      PDF

   2. Introduction

      • Parisa Hariri, Riku Klén, Matti Vuorinen

      Pages 3-5

   3. A Survey of Quasiregular Mappings

      • Parisa Hariri, Riku Klén, Matti Vuorinen

      Pages 7-21

3. Part II

   1. Front Matter

      Pages 23-23

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   2. Möbius Transformations

      • Parisa Hariri, Riku Klén, Matti Vuorinen

      Pages 25-48

   3. Hyperbolic Geometry

      • Parisa Hariri, Riku Klén, Matti Vuorinen

      Pages 49-66

   4. Generalized Hyperbolic Geometries

      • Parisa Hariri, Riku Klén, Matti Vuorinen

      Pages 67-81

   5. Metrics and Geometry

      • Parisa Hariri, Riku Klén, Matti Vuorinen

      Pages 83-100

4. Part III

   1. Front Matter

      Pages 101-102

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   2. The Modulus of a Curve Family

      • Parisa Hariri, Riku Klén, Matti Vuorinen

      Pages 103-131

   3. The Modulus as a Set Function

      • Parisa Hariri, Riku Klén, Matti Vuorinen

      Pages 133-147

   4. The Capacity of a Condenser

      • Parisa Hariri, Riku Klén, Matti Vuorinen

      Pages 149-172

   5. Conformal Invariants

      • Parisa Hariri, Riku Klén, Matti Vuorinen

      Pages 173-187

5. Part IV

   1. Front Matter

      Pages 189-190

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   2. Hyperbolic Type Metrics

      • Parisa Hariri, Riku Klén, Matti Vuorinen

      Pages 191-208

   3. Comparison of Metrics

      • Parisa Hariri, Riku Klén, Matti Vuorinen

      Pages 209-238

   4. Local Convexity of Balls

      • Parisa Hariri, Riku Klén, Matti Vuorinen

      Pages 239-259

   5. Inclusion Results for Balls

      • Parisa Hariri, Riku Klén, Matti Vuorinen

      Pages 261-278

6. Part V

   1. Front Matter

      Pages 279-279

      PDF

This book is an introduction to the theory of quasiconformal and quasiregular mappings in the euclidean n-dimensional space, (where n is greater than 2). There are many ways to develop this theory as the literature shows. The authors' approach is based on the use of metrics, in particular conformally invariant metrics, which will have a key role throughout the whole book. The intended readership consists of mathematicians from beginning graduate students to researchers. The prerequisite requirements are modest: only some familiarity with basic ideas of real and complex analysis is expected.

• Möbius transformations
• Quasiconformal mappings in the plane
• Quasiregular mappings in Rn
• conformal invariants
• boundary properties of QR-maps
• hyperbolic metric

“The book not only provides a reference for the study of quasiregular mappings, but could also serve as a useful handbook for the student/researcher interested in hyperbolic (and hyperbolic-type) metrics on Euclidean domains. … it constitutes a significant addition to the body of literature on these topics.” (David Matthew Freeman, Mathematical Reviews, February, 2022)

• Department of Mathematics and Statistics, University of Turku, Turku, Finland

  Parisa Hariri, Matti Vuorinen

• Turku PET Centre, University of Turku, Turku, Finland

  Riku Klén

Matti Vuorinen, currently professor of mathematics at the University of Turku and docent at the University of Helsinki, is the author of more than 200 publications, including 2 books on quasiregular and quasiconformal mappings. The first entitled "Conformal geometry and quasiregular mappings" (Lecture Notes in Math. Vol. 1319) was published by Springer-Verlag in 1988 and the second, entitled "Conformal invariants, inequalities and quasiconformal mappings" by J. Wiley, in 1997.



Riku Klén, currently assistant professor at the University of Turku, Turku PET Centre, does research in Conformal Geometry and Quasiconformal Mappings as well as Medical Imaging.

Parisa Hariri, obtained her PhD in Mathematics from the University of Turku in 2018, under the supervision of Matti Vuorinen and Riku Klen. Her PhD thesis was on 'Hyperbolic Type Metrics in Geometric Function Theory'. She is currently working as medical statistician at the University of Oxford Vaccine Group in the Department of Paediatrics.

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