# What is “Teichmüller Theory” and its history?

What is "Teichmüller Theory"? What part has been worked out / foreseen 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?

• 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, Teichmüller theory, vol. 1. Let me try to list briefly Teichmüller's own contribution to Teichmüller theory. Bers's papers of 1960-s are good primary sources. The few papers of Teichmüller 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 the introduction of Teichmüller's space (instead of the much more complicated moduli space). It is simply connected! The second main contribution is the definition of the Teichmüller 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 Grötzsch. Teichmüller's contribution was a) considering them on compact Riemann surfaces, and b) describing the extremal map in terms of a certain quadratic differential. He also established existence and uniqueness of the extremal mapping with a very original argument.

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

Teichmüller died young (he was killed or MIA in the Eastern front, somewhere near Kiev in 1943), 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 Teichmüller 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 cotangent space as a space of quadratic differentials. Later Royden proved that the Teichmüller distance coincides with the Kobayashi distance.

The crucial technical tool, existence and analytic dependence on parameters of the 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 in Teichmüller's time. It was published for the first time by Boyarski in 1955.

This more or less constitutes the original "Teichmüller theory". Later the meaning of the term substantially expanded, to include almost everything about the moduli spaces.

EDIT. The good news is that all principal papers of Teichmüller are now available in English:

MR3560242 Handbook of Teichmüller theory. Vols. IV,V,VI. European Mathematical Society (EMS), Zürich, 2016. (Each volume contains translations of several papers of Teichmüller)

• @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