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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 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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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 ball domain in $C^{3g-3}$. C^{3g-3}$which is homeomorphic to$R^{6g-6}$. They identified the contangent space as a space of quadratic differentials. The crucial technical tool, existence-and-analytic-dependence-of-parameters-of homeomorphic solution of Beltrami equation with$L^\infty$coefficient, which people call sometimes the "measurable Riemann theorem" was not available at Teichmuller's time. It was published for the first time by Boyarski in 1955. This more or less constitutes the original "Teichmuller theory". Later the meaning of the term substantially expanded, to include almost everything about the moduli spaces. 1 First of all, let me recommend a book: J. Hubbard, Teichmuller theory, vol. 1. Let me try to list briefly Teichmuller's own contribution to Teichmuller theory. Bers's papers of 1960-s are a good primary source. Few papers of Teichmuller himself that I read are also exciting but my poor knowledge of German does not allow me to read all of them. Perhaps the main contribution is introduction of Teichmuller's space (instead of the much more complicated Moduli space). It is simply connected! Second main contribution is the definition of Teichmuller metric on this space. The metric is defined using a solution of an extremal problem: finding a quasiconformal homeomorphism in a homotopy class with smallest dilatation. Such problems in plane domains were first considered by Grotsch, Teichmuller's contribution was a) considering them on compact Riemann surfaces, and b) describing the extremal map in terms of certain quadratic differential. He also established existence and uniqueness of the extremal mapping with a very original argument. Teichmuller distance is defined as$(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$is isomorphic to a ball in$C^{3g-3}$. They identified the contangent space as a space of quadratic differentials. The crucial technical tool, existence-and-analytic-dependence-of-parameters-of homeomorphic solution of Beltrami equation with$L^\infty\$ coefficient, 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.