Conformally Invariant Metrics and Quasiconformal Mappings

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.

Part I: Introduction and Review.- Introduction.- A Survey of QuasiregularMappings.- Part II: Conformal Geometry.- Möbius Transformations.- Hyperbolic Geometry.- Generalized Hyperbolic Geometries.- Metrics and Geometry.- Part III: Modulus and Capacity.- The Modulus of a Curve Family.- The Modulus as a Set Function.- The Capacity of a Condenser.- Conformal Invariants.- Part IV: Intrinsic Geometry.- Hyperbolic Type Metrics.- Comparison of Metrics.- Local Convexity of Balls.- Inclusion Results for Balls.- Part V: QuasiregularMappings.- Basic Properties of QuasiregularMappings.- Distortion Theory.- Dimension-Free Theory.- Metrics and Maps.- Teichmüller's Displacement Problem.- Part VI: Solutions.- Solutions to Exercises.

"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)

“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)