EAN: 9783662470091

COLLECTED WORKS IN MATHEMATICS GESAMMELTE ABH...

During his short life Oswald Teichmüller wrote 34 papers  all reproduced in this volume. From
the Preface: Teichmüller's most influential paper was called Extremale quasikonforme
Abbildungen und quadratische Differentiale (No. 20 in this collection). At the time of its
appearance several special cases of extremal problems for quasiconformal mappings had already
been solved  and Teichmüller was able to draw on a substantial fund of experience. Nevertheless
it was a remarkable feat to extract the common features of all the known examples and formulate
a conjecture  now known as Teichmüller's theorem  which in an unexpected way connects the
holomorphic second order differentials on a Riemann surface with the extremal quasiconformal
mappings of that surface. The paper of 1939 contains a uniqueness proof  which is essentially
still the only known proof  but not yet a rigorous existence proof. This did not prevent
Teichmüller from laying the foundation of what has become known as the theory of Teichmüller
spaces  a theory that has mushroomed to an extent that could not then have been foreseen. At
the same time Teichmüller's work led to a deeper understanding of the fundamental role played
by quasiconformal mappings in all of geometric function theory  and it foreshadowed the
subsequent development of the theory of quasiconformal mappings in several dimensions... the
whole theory of analytic functions of one complex variable has been greatly enriched by the
inclusion of quasiconformal mappings  much of it based on Teichmüller's seminal ideas.

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