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. The paper of 1939 contains a uniqueness proof, which is essentially still the only known proof, but not yet a rigorous existence proof. 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. 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. From the Preface: "Teichmüller's most influential paper was called "Extremale quasikonforme Abbildungen und quadratische Differentiale" (No. During his short life Oswald Teichmüller wrote 34 papers, all reproduced in this volume.
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