What is "Teichmüller Theory" and its history ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T07:09:26Z http://mathoverflow.net/feeds/question/114787 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/114787/what-is-teichmuller-theory-and-its-history What is "Teichmüller Theory" and its history ? Alexander Chervov 2012-11-28T17:01:20Z 2012-11-28T19:45:51Z <p>What is "Teichmüller Theory" ? What part has been worked out / forseen by O. Teichmüller himself and what is further development ? Is there some current work which might be considered as continuation/completion of this theory ? </p> <p><strong>Background</strong> The question might be seen as too naive and can be answered by google or <a href="http://en.wikipedia.org/wiki/Teichm%C3%BCller_space" rel="nofollow">Wikipedia</a>, but I have it in mind for a long time and do not think that it is that much simple. Let me explain what is puzzling me: Teichmüller space is very close to moduli space of Riemann surfaces ("The Teichmüller space is the universal covering orbifold of the (Riemann) moduli space.") and reading some sources make me expression that "Teichmüller Theory" is everything which is related to the moduli space of Riemann surfaces. Is it really like this ? If it is true it does not seem to me good name since "theory" should be something not so diverse as current research on moduli spaces of Riemann surfaces.</p> <p>On the other hand what I heard about the contribution of Teichmüller himself - it is introduction of the Teichmüller metric by means of quasiconformal maps. (<a href="http://en.wikipedia.org/wiki/Teichm%C3%BCller_metric#Teichm.C3.BCller_metric" rel="nofollow">See Wikipedia</a>). It is beautiful result, but it is kind of "theorem", not "theory", so probably there is something more ? which I am missing ?</p> <p>The question might be considered as background to <a href="http://mathoverflow.net/questions/114676/what-are-some-applications-of-teichmuller-theory" rel="nofollow">http://mathoverflow.net/questions/114676/what-are-some-applications-of-teichmuller-theory</a> </p> http://mathoverflow.net/questions/114787/what-is-teichmuller-theory-and-its-history/114792#114792 Answer by Alexandre Eremenko for What is "Teichmüller Theory" and its history ? Alexandre Eremenko 2012-11-28T17:40:40Z 2012-11-28T19:45:51Z <p>First of all, let me recommend a book: J. Hubbard, Teichmuller theory, vol. 1. Let me try to list briefly Teichmuller's own contribution to Teichmuller theory. Bers's papers of 1960-s are a good primary source. Few papers of Teichmuller himself that I read are also exciting but my poor knowledge of German does not allow me to read all of them. </p> <p>Perhaps the main contribution is introduction of Teichmuller's space (instead of the much more complicated Moduli space). It is simply connected! Second main contribution is the definition of Teichmuller metric on this space. The metric is defined using a solution of an extremal problem: finding a quasiconformal homeomorphism in a homotopy class with smallest dilatation. Such problems in plane domains were first considered by Grotsch, Teichmuller's contribution was a) considering them on compact Riemann surfaces, and b) describing the extremal map in terms of certain quadratic differential. He also established existence and uniqueness of the extremal mapping with a very original argument.</p> <p>Teichmuller distance is defined as $(1/2)\log K$, where $K$ is the extremal dilatation. </p> <p>Teichmuller died young (he was killed or MIA in the Eastern front, somewhere near Kiev in 1944), and many of his principal papers contain a lot of heuristic arguments.</p> <p>The subject was developed by Ahlfors and Bers in 1950-s. They rigorously introduced the analytic structure on Teichmuller spaces, and proved in particular that the Teichmuller space of surfaces of genus $g>1$ is isomorphic to a domain in $C^{3g-3}$ which is homeomorphic to $R^{6g-6}$. They identified the contangent space as a space of quadratic differentials. Later Royden proved that the Teichmuller distance coincides with the Kobayashi distance.</p> <p>The crucial technical tool, existence-and-analytic-dependence-of-parameters-of homeomorphic solution of the Beltrami equation with $L^\infty$ norm of the coefficient less than 1, which people call sometimes the "Measurable Riemann theorem" was not available at Teichmuller's time. It was published for the first time by Boyarski in 1955.</p> <p>This more or less constitutes the original "Teichmuller theory". Later the meaning of the term substantially expanded, to include almost everything about the moduli spaces. </p>