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Infinitesimal Geometry of Quasiconformal and Bi-Lipschitz Mappings in the Plane
EMS Tracts in Mathematics Vol. 19

Bogdan Bojarski (IM PAN, Warsaw, Poland)
Vladimir Gutlyanskii (National Academy of Science of Ukraine, Donetsk, Ukraine)
Olli Martio (Finnish Academy of Science and Letters, Helsinki, Finland)
Vladimir Ryazanov (National Academy of Science of Ukraine, Donetsk, Ukraine)

Infinitesimal Geometry of Quasiconformal and Bi-Lipschitz Mappings in the Plane

ISBN print 978-3-03719-122-4, ISBN online 978-3-03719-622-9
DOI 10.4171/122
May 2013, 214 pages, hardcover, 17 x 24 cm.
58.00 Euro

This book is intended for researchers interested in new aspects of local behavior of plane mappings and their applications. The presentation is self-contained, but the reader is assumed to know basic complex and real analysis.

The study of the local and boundary behavior of quasiconformal and bi-Lipschitz mappings in the plane forms the core of the book. The concept of the infinitesimal space is used to investigate the behavior of a mapping at points without differentiability. This concept, based on compactness properties, is applied to regularity problems of quasiconformal mappings and quasiconformal curves, boundary behavior, weak and asymptotic conformality, local winding properties, variation of quasiconformal mappings, and criteria of univalence. Quasiconformal and bi-Lipschitz mappings are instrumental for understanding elasticity, control theory and tomography and the book also offers a new look at the classical areas such as the boundary regularity of a conformal map. Complicated local behavior is illustrated by many examples.

The text offers a detailed development of the background for graduate students and researchers. Starting with the classical methods to study quasiconformal mappings, this treatment advances to the concept of the infinitesimal space and then relates it to other regularity properties of mappings in Part II. The new unexpected connections between quasiconformal and bi-Lipschitz mappings are treated in Part III. There is an extensive bibliography.

Keywords: Quasiconformal mappings, bi-Lipschitz mappings, Beltrami equations, local and boundary behavior, infinitesimal space, convergence and compactness theory, asymptotic linearity, rotation problems, conformal differentiability

Further Information

Review in MR 3088451