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This volume is a collection of surveys on function theory in
euclidean n-dimensional spaces centered around the theme of
quasiconformal space mappings. These surveys cover or are related
to several topics including inequalities for conformal invariants
and extremal length, distortion theorems, L(p)-theory of
quasiconformal maps, nonlinear potential theory, variational
calculus, value distribution theory of quasiregular maps,
topological properties of discrete open mappings, the action of
quasiconformal maps in special classes of domains, and global
injectivity theorems. The present volume is the first collection of
surveys on Quasiconformal Space Mappings since the origin of the
theory in 1960 and this collection provides in compact form access
to a wide spectrum of recent results due to well-known specialists.
CONTENTS: G.D. Anderson, M.K. Vamanamurthy, M. Vuorinen: Conformal
invariants, quasiconformal maps and special functions.- F.W.
Gehring: Topics in quasiconformal mappings.- T.Iwaniec: L(p)-theory
of quasiregular mappings.- O. Martio: Partial differential
equations and quasiregular mappings.- Yu.G. Reshetnyak: On
functional classes invariant relative to homothetics.- S. Rickman:
Picard's theorem and defect relation for quasiconformal mappings.-
U. Srebro: Topological properties of quasiregular mappings.- J.
V{is{l{: Domains and maps.- V.A. Zorich: The global homeomorphism
theorem for space quasiconformal mappings, its development and
related open problems.
This book is an introduction to the theory of spatial quasiregular
mappings intended for the uninitiated reader. At the same time the
book also addresses specialists in classical analysis and, in
particular, geometric function theory. The text leads the reader to
the frontier of current research and covers some most recent
developments in the subject, previously scatterd through the
literature. A major role in this monograph is played by certain
conformal invariants which are solutions of extremal problems
related to extremal lengths of curve families. These invariants are
then applied to prove sharp distortion theorems for quasiregular
mappings. One of these extremal problems of conformal geometry
generalizes a classical two-dimensional problem of O.
TeichmA1/4ller. The novel feature of the exposition is the way in
which conformal invariants are applied and the sharp results
obtained should be of considerable interest even in the
two-dimensional particular case. This book combines the features of
a textbook and of a research monograph: it is the first
introduction to the subject available in English, contains nearly a
hundred exercises, a survey of the subject as well as an extensive
bibliography and, finally, a list of open problems.
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.
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