# What is “Teichmüller Theory” and its history ?

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 ?

Background The question might be seen as too naive and can be answered by google or Wikipedia, 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.

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. (See Wikipedia). It is beautiful result, but it is kind of "theorem", not "theory", so probably there is something more ? which I am missing ?

The question might be considered as background to What are some Applications of Teichmüller Theory?

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This should be community wiki. –  Igor Rivin Nov 28 '12 at 17:33
@Igor why? There is no unique answer? How can it be? –  Alexander Chervov Nov 28 '12 at 17:41
@Alexander: Teichmuller theory is a large subject, with a long and complicated history, so there is no unique answer... –  Igor Rivin Nov 28 '12 at 20:29
Sasha, I would suggest the (somewhat passionate) review of W.Abikoff for Nag's book "The complex analytic theory of Teichmueller spaces": projecteuclid.org/… –  Peter Dalakov Nov 28 '12 at 21:48
@Igor well, 1) representation theory is also big, but we can characterize it in one sentence as studies of linear representation of groups, algebras, etc... I am missing even such oversimplified characterization, and it is seems not only I, but e.g. Wikipedia also. 2) Concerning history - in the body of the question I tried to narrow down the question - what are Teichmuller's own contribution ? PS Probably you are the person who can say a lot about Teichmuller theory, may be you can share your knowledge and then it would be also evident for me that question should be CW... –  Alexander Chervov Nov 29 '12 at 6:10

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.

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.

Teichmuller distance is defined as $(1/2)\log K$, where $K$ is the extremal dilatation.

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.

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.

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.

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.

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Thank you very much ! May I ask what was the motivation for Teichmuller ? Did his solve some previously known problem or just observed that space bearing his name now is something beautiful and begin its study ? –  Alexander Chervov Nov 28 '12 at 17:59
As I said, Teichmuler wrote in German, and I read very little of his work. So my opinions are mostly based on the papers of Ahlfors and Bers. To my understanding, he wanted to lay a rigorous foundation for the moduli spaces. Moduli spaces were used since Riemann, who obtained this number 6g-6 by parameter count. But they were never defined rigorously. Teichmuller had an insight that the extremal problems considered by Grotsch can be used to understand the "space of Riemann surfaces", and the crucial idea to consider Teichmuller space instead of the moduli space. –  Alexandre Eremenko Nov 28 '12 at 19:39
@Alexandre: I would add here that "Teichmuller space" in some form was introduced already in the work of Klein, Fricke, Fenchel and Nielsen, which preceded the one of Teichmuller. Namely, they were working with the space of discrete embeddings of closed surface groups to $PSL(2,R)$ (which is naturally homeomorphic to the Teichmuller space by the uniformization theorem) and already knew that it is homeomorphic to $R^{6g-6}$ (Fenchel-Nielsen coordinates). What they did not have was a natural complex structure (invariant under the mapping class group). –  Misha Nov 28 '12 at 20:15
Misha, thanks. This makes the thing clearer. Unfortunately my German is too poor to read Fricke, Klein and Teichmuller, so my knowledge is based on Ahlfors and Bers, and they were not very accurate with history and references:-) –  Alexandre Eremenko Dec 6 '13 at 15:04