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Teichmüller space
In mathematics, the Teichmüller space TX of a (real) topological surface X, is a space that parameterizes complex structures on X up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Each point in TX may be regarded as an isomorphism class of 'marked' Riemann surfaces where a 'marking' is an isotopy class of homeomorphisms from X to X. The Teichmüller space is the universal covering orbifold of the (Riemann) moduli space.
Teichmüller space has a canonical complex manifold structure and a wealth of natural metrics. The underlying topological space of Teichmüller space was studied by Fricke, and the Teichmüller metric on it was introduced by Oswald Teichmüller (1939).
Complex structures and Riemann surfaces
Each topological atlas for a (real) surface X consists of injective maps from open subsets of X into the Euclidean plane. Identify the Euclidean plane with the complex plane via \( (x,y) \mapsto x+ y \sqrt{-1} \) . A topological atlas is a complex atlas for X if each transition map is a biholomorphism. Two complex atlases are equivalent provided their union is a complex atlas. An equivalence class of complex atlases is called a complex structure. A topological surface X equipped with a complex structure is called a Riemann surface. Among all atlases belonging to a complex structure, there is a maximal atlas which is the union of all complex atlases in the complex structure. One may identify each complex structure with this maximal atlas.
Teichmüller space as the set of equivalence classes of complex structures
Given two complex structures on X, let \(\{\varphi: U \rightarrow {\mathbf R}^2\} \) and \( \{\psi: V \rightarrow {\mathbf R}^2\} \) be the associated maximal atlases. The two complex structures are said to be Teichmüller equivalent provided there exists a homeomorphism \( f: X \rightarrow X \) that is isotopic to the identity homeomorphism so that \(\{\varphi \circ f\} = \{\psi \} \) . The Teichmüller space TX is defined to be the set of Teichmüller equivalence classes of complex structures on X.
Relation to the moduli space of Riemann surfaces
In the definition of Teichmüller equivalence, the homeomorphism f is required to be isotopic to the identity homeomorphism. If this requirement is dropped, then we obtain a new equivalence relation whose equivalence classes form the Riemann moduli space of X. In particular, if two complex structures on X differ by a homeomorphism, then they define the same point in moduli space. Yet, if the homeomorphism is not isotopic to the identity homeomorphism, then the two complex structures define different points in Teichmüller space. In sum, each point of Teichmüller space contains additional information. This additional information is called a marking and may be regarded as an isotopy class of homeomorphisms\( f: X \rightarrow X \) . Forgetting the marking defines a map from Teichmüller space to moduli space which is a universal orbifold covering map.
The action of the group of homeomorphisms
Both Teichmüller space and the Riemann moduli space may be more concisely defined in terms of a group action. The set of all homeomorphisms \(f:X\rightarrow X \) underlies the group \({\rm Homeo}(X) \) whose binary operation is composition. The assignment \((\varphi, f) \mapsto \varphi \circ f \) is a group action on the set of complex structures. The Riemann moduli space of X is the orbit space of this action. The homeomorphisms that are isotopic to the identity homeomorphism constitute a subgroup \( {\rm Homeo}_0(X) of {\rm Homeo}(X) \) . This subgroup also acts on the set of complex structures, and the resulting orbit space is the Teichmüller space.
Relation to the mapping class group
The group \( {\rm Homeo}_0(X) \) is a normal subgroup of \( {\rm Homeo}(X) \) . The quotient group is called the mapping class group of X. The elements of this group are isotopy classes of homeomorphisms of X or mapping classes. The mapping class group acts on Teichmüller space and the resulting orbit space is the Riemann moduli space.
Properties of TX
The Teichmüller space of X is a complex manifold. Its complex dimension depends on topological properties of X. If X is obtained from a compact surface of genus g by removing n points, then the dimension of TX is 3g − 3 + n whenever this number is positive. These are the cases of "finite type". In these cases, it is homeomorphic to a complex vector space of this dimension, and in particular is contractible.
Note that, even though a compact surface with a point removed and the same surface with a disc removed are topologically the same, a complex structure on the surface behaves very differently around a point and around a removed disc. In particular, the boundary of the removed disc becomes an "ideal boundary" for the Riemann surface, and isomorphisms between surfaces with non-empty ideal boundary must take this ideal boundary into account. Varying the structure quasiconformally along the ideal boundary shows that the Teichmüller space of a Riemann surface with nonempty ideal boundary must be infinite-dimensional.
Metrics on Teichmüller space
Teichmüller space has a large number of different natural metrics. Here are some of the more commonly used ones, with the most important ones first.
Teichmüller metric
There is, in general, no isomorphism from one Riemann surface to another of the same topological type that is isotopic to the identity. In the case of surfaces of finite type, there is, however, always a quasiconformal map from one to the other that is isotopic to the identity. Between any two such Riemann surfaces there is an extremal quasiconformal map called the Teichmüller mapping whose maximal quasiconformal dilation K is as small as possible, and log K gives a metric on TX, called the Teichmüller metric.
The Teichmüller metric is a complete Finsler metric, but is not usually Riemannian. Any two points are joined by a unique geodesic. Masur showed that there are two geodesics such that their distance function is bounded, and in particular not convex, contradicting an earlier published claim.
Weil–Petersson metric
The Weil–Petersson metric is a Riemannian metric on Teichmüller space. Ahlfors showed that it is a Kähler metric. It is not complete in general.
Thurston’s asymmetric metric
This is not a metric in the usual sense as it is not symmetric. It was introduced by Thurston (1998). The papers Papadopoulos (1991) Théret (2007) Théret (2008) contain results about the geodesics of this metric.
Bergman metric
This is a special case of the Bergman metric on any domain of holomorphy.
Carathéodory metric
This is a special case of the Carathéodory metric of any complex space.
Kähler–Einstein metric
Cheng and Yau showed that there is a unique complete Kähler–Einstein metric on Teichmüller space. It has constant negative scalar curvature.
Kobayashi metric
This is a special case of the Kobayashi metric defined on any complex space. Royden (1970) showed that it coincides with the Teichmüller metric.
McMullen metric
This is a complete Kähler metric of bounded sectional curvature introduced by McMullen (2000) that is Kähler-hyperbolic.
Compactifications of Teichmüller spaces
There are several inequivalent compactifications of Teichmüller spaces that have been studied. Several of the earlier compactifications depend on the choice of a point in Teichmüller space so are not invariant under the modular group, which can be inconvenient. Thurston later found a compactification without this disadvantage, which has become the most widely used compactification.
Bers compactification
The Bers compactification is given by taking the closure of the image of the Bers embedding of Teichmüller space, studied by Lipman Bers (1970). The Bers embedding depends on the choice of a point in Teichmüller space so is not invariant under the modular group, and in fact the modular group does not act continuously on the Bers compactification.
Teichmüller compactification
The "points at infinity" in the Teichmüller compactification consist of geodesic rays (for the Teichmüller metric) starting at a fixed basepoint. This compactification depends on the choice of basepoint so is not acted on by the modular group, and in fact Kerckhoff showed that the action of the modular group on Teichmüller space does not extend to a continuous action on this compactification.
Thurston compactification
By looking at the hyperbolic lengths of simple closed curves for each point in Teichmüller space and taking the closure in the (infinite-dimensional) projective space, Thurston (1988) introduced a compactification whose points at infinity correspond to projective measured laminations. The compactified space is homeomorphic to a closed ball. This Thurston compactification is acted on continuously by the modular group. In particular any element of the modular group has a fixed point in Thurston's compactification, which Thurston used in his classification of elements of the modular group.
Gardiner-Masur compactification
Gardiner & Masur (1991) considered a compactification similar to the Thurston compactification, but using extremal length rather than hyperbolic length. The modular group acts continuously on this compactification, but they showed that their compactification has strictly more points at infinity.
Examples of Teichmüller spaces
The Teichmüller spaces T0,0, T0,1, T0,2, T0,3 (corresponding to a sphere with at most 3 points removed) are points.
The Teichmüller spaces T0,4, T1,0, T1,1, corresponding to the sphere with four points removed, the torus, and the torus with one point removed all have isomorphic Teichmüller spaces, which can be identified with the complex upper half plane.
References
Bers, Lipman (1970), "On boundaries of Teichmüller spaces and on Kleinian groups. I", Annals of Mathematics. Second Series 91: 570–600, JSTOR 1970638, MR 0297992
Bers, Lipman (1981), "Finite-dimensional Teichmüller spaces and generalizations", American Mathematical Society. Bulletin. New Series 5 (2): 131–172, doi:10.1090/S0273-0979-1981-14933-8, MR 621883
Gardiner, Frederick P. (1987), Teichmüller theory and quadratic differentials, Pure and Applied Mathematics (New York), New York: John Wiley & Sons, ISBN 978-0-471-84539-3, MR 903027
Gardiner, Frederic P.; Masur, Howard (1991), "Extremal length geometry of Teichmüller space", Complex Variables Theory Appl. 16 (2–3): 209–237, MR 1099913
Hubbard, John Hamal (2006), Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 1, Matrix Editions, Ithaca, NY, ISBN 978-0-9715766-2-9, MR 2245223
McMullen, Curtis T. (2000), "The moduli space of Riemann surfaces is Kähler hyperbolic", Annals of Mathematics. Second Series 151 (1): 327–357, doi:10.2307/121120, MR 1745010
Papadopoulos, Athanase (1991), "On Thurston's boundary of Teichmüller space and the extension of earthquakes", Topology and its Applications 41 (2): 147–177, doi:10.1016/0166-8641(91)90001-3, MR 1135095
Papadopoulos, Athanase, ed. (2007), Handbook of Teichmüller theory. Vol. I, IRMA Lectures in Mathematics and Theoretical Physics 11, European Mathematical Society (EMS), Zürich, doi:10.4171/029, ISBN 978-3-03719-029-6, MR 2284826
Papadopoulos, Athanase, ed. (2009), Handbook of Teichmüller theory. Vol. II, IRMA Lectures in Mathematics and Theoretical Physics 13, European Mathematical Society (EMS), Zürich, doi:10.4171/055, ISBN 978-3-03719-055-5, MR 2524085
Papadopoulos, Athanase, ed. (2012), Handbook of Teichmüller theory. Vol. III, IRMA Lectures in Mathematics and Theoretical Physics 17, European Mathematical Society (EMS), Zürich, doi:10.4171/103, ISBN 978-3-03719-103-3
Papadopoulos, Athanase, ed. (2014), Handbook of Teichmüller theory. Vol. IV, IRMA Lectures in Mathematics and Theoretical Physics 19, European Mathematical Society (EMS), Zürich, doi:10.4171/117, ISBN 978-3-03719-117-0
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