Vasil#ev A. Moduli of families of curves for conformal and quasiconformal mappings (LNM1788, Springer, 2002)(217s)_MCc_.pdf

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Lecture Notes in Mathematics
Editors:
J.-M. Morel, Cachan
F. Takens, Groningen
B. Teissier, Paris
1788
3
Berlin
Heidelberg
New York
Barcelona
Hong Kong
London
Milan
Paris
Tokyo
Alexander Vasil’ev
Moduli of Families of Curves
for Conformal and
Quasiconformal Mappings
13
Author
Alexander Vasil’ev
Departamento de Matem´ tica
a
Universidad T´cnica Federico Santa Mar´a
e
ı
Casilla 110-V, Valpara´so, Chile
ı
E-mail: alexander.vasiliev@mat.utfsm.cl
Cataloging-in-Publication Data applied for
Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Vasil'ev, Aleksandr:
Moduli of families of curves for conformal and quasiconformal mappings /
Alexander Vasil'ev. - Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong
; London ; Milan ; Paris ; Tokyo : Springer, 2002
(Lecture notes in mathematics ; 1788)
ISBN 3-540-43846-7
Mathematics Subject Classification (2000):
30C35, 30C55, 30C62, 30C75, 30F10, 30F60
ISSN
0075-8434
ISBN
3-540-43846-7
Springer-Verlag Berlin Heidelberg New York
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© Springer-Verlag Berlin Heidelberg
2002
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SPIN:
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Preface
In the present monograph, we consider the extremal length method in its form
of
the method of moduli of families of curves
in applications to the problems
of conformal, quasiconformal mapping, and Teichm¨ller spaces. This method
u
going back to H. Gr¨tzsch, A. Beurling, L. V. Ahlfors, J. Jenkins is now
o
one of the basic methods in various parts of Analysis. Several surveys and
monographs, e.g., [30], [64], [78], [107], [139] are devoted to the development
of this method and applications. However, we want to give here a useful guide:
how one can start to solve extremal problems of conformal mapping beginning
with simple but famous classical theorems and ending at difficult new results.
Some more non-traditional applications we consider in the quasiconformal
case. The modulus method permits us to consider the problems in question
from a single point of view.
At the mid-century it was established that the classical methods of the
geometric function theory could be extended to complex hyperbolic mani-
folds. The Teichm¨ller spaces turned out to be the most important of them.
u
Recently, it has become clear that some forms of the extremal length method
could be applied to examine different properties of Teichm¨ller spaces (see
u
e.g. [43], [44]).
Thus, we are concerned with
the modulus method and its applications to
extremal problems for conformal, quasiconformal mappings, the extension of
moduli onto Teichm¨ller spaces.
The book is intended for different groups of
u
readers:
(1) Non-experts who want to know about how one can use the modulus
technique to solve extremal problems of Complex Analysis. One can find
proofs of classical theorems of conformal and quasiconformal mapping by
means of the modulus method as well as many examples of symmetrization
and polarization. Graduate students will find here some useful exercises to
check their understanding.
(2) Experts who will find new results about solution of difficult extremal prob-
lems for conformal and quasiconformal mappings and about the extension of
the modulus onto Teichm¨ller spaces.
u

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