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Gesammelte Abhandlungen - Collected Papers (English, German, Paperback, Reprint 2015 of the 1982 edition)
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Gesammelte Abhandlungen - Collected Papers (English, German, Paperback, Reprint 2015 of the 1982 edition)
Series: Springer Collected Works in Mathematics
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During his short life Oswald Teichmuller wrote 34 papers, all
reproduced in this volume. From the Preface: "Teichmuller'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
Teichmuller 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 Teichmuller'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 Teichmuller from laying the foundation of what
has become known as the theory of Teichmuller spaces, a theory that
has mushroomed to an extent that could not then have been foreseen.
At the same time Teichmuller'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 Teichmuller's seminal
ideas."
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