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First of all, let me recommend a book: J. Hubbard, Teichmuller theory, vol. 1. Let me try to list briefly Teichmüller's own contribution to Teichmuller 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 Teichmuller's space (instead of the much more complicated moduli space). It is simply connected! The second main contribution is the definition of the 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 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.

Teichmuller 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 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 cotangent 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 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 "Teichmuller 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 Teichmuller 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 Teichmuller)

And with comments.

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)

And with comments.

First of all, let me recommend a book: J. Hubbard, Teichmuller theory, vol. 1. Let me try to list briefly Teichmüller's own contribution to Teichmuller 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 Teichmuller's space (instead of the much more complicated moduli space). It is simply connected! The second main contribution is the definition of the 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 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.

Teichmuller 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 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 cotangent 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 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 "Teichmuller 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 Teichmuller 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 Teichmuller)

And with comments.

First of all, let me recommend a book: J. Hubbard, Teichmuller theory, vol. 1. Let me try to list briefly Teichmüller's own contribution to Teichmuller 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 Teichmuller's space (instead of the much more complicated moduli space). It is simply connected! The second main contribution is the definition of the 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 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.

Teichmuller 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 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 cotangent 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 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 "Teichmuller 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 Teichmuller 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 Teichmuller)

And with comments.

First of all, let me recommend a book: J. Hubbard, Teichmuller theory, vol. 1. Let me try to list briefly Teichmüller's own contribution to Teichmuller 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 Teichmuller's space (instead of the much more complicated moduli space). It is simply connected! The second main contribution is the definition of the 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 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.

Teichmuller 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 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 cotangent 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 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 "Teichmuller theory". Later the meaning of the term substantially expanded, to include almost everything about the moduli spaces.

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

MR3560242 Handbook of Teichmüller theory. Vol. VI. European Mathematical Society (EMS), Zürich, 2016.

And with comments.

First of all, let me recommend a book: J. Hubbard, Teichmuller theory, vol. 1. Let me try to list briefly Teichmüller's own contribution to Teichmuller 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 Teichmuller's space (instead of the much more complicated moduli space). It is simply connected! The second main contribution is the definition of the 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 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.

Teichmuller 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 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 cotangent 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 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 "Teichmuller 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 Teichmuller 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 Teichmuller)

And with comments.