In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Teichmüller space T_X of a (real) topological surface X, is a space that parameterizes complex structures

Complex manifold

In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....

 on X up to the action of homeomorphism

Homeomorphism

In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...

s that are isotopic to the identity homeomorphism

Identity function

In mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument...

. Each point in T_X may be regarded as an isomorphism class of 'marked' Riemann surface

Riemann surface

In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...

s where a 'marking' is an isotopy class of homeomorphisms from X to X.
The Teichmüller space is the universal covering orbifold

Orbifold

In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold is a generalization of a manifold...

 of the (Riemann) moduli space.

Teichmüller space has a canonical complex

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

 manifold

Manifold

In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the 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 .

Complex structures and Riemann surfaces

Each topological atlas

Atlas (topology)

In mathematics, particularly topology, one describesa manifold using an atlas. An atlas consists of individualcharts that, roughly speaking, describe individual regionsof the manifold. If the manifold is the surface of the Earth,...

 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

Complex plane

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...

 via . A topological atlas is a complex atlas for X if each transition function

Transition function

In mathematics, a transition function has several different meanings:* In topology, a transition function is a homeomorphism from one coordinate chart to another...

 is a biholomorphism. Two complex atlases are equivalent

Equivalence relation

In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...

 provided their union

Union (set theory)

In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...

 is a complex atlas. An equivalence class of complex atlases is called a complex structure

Complex manifold

In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....

. A topological surface X equipped with a complex structure is called a Riemann surface

Riemann surface

In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...

. Among all atlases belonging to a complex structure, there is a maximal atlas

Atlas (topology)

In mathematics, particularly topology, one describesa manifold using an atlas. An atlas consists of individualcharts that, roughly speaking, describe individual regionsof the manifold. If the manifold is the surface of the Earth,...

 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 and be the
associated maximal atlases. The two complex structures are said to be Teichmüller equivalent provided there exists a homeomorphism

Homeomorphism

In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...

 
that is isotopic to the identity homeomorphism

Identity function

In mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument...

 so that . The Teichmüller space T_X 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 is required to be isotopic to the identity homeomorphism. If we drop this requirement, then we obatin 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 . Forgetting the marking defines a map from Teichmüller space to moduli space which is an universal orbifold covering map

Orbifold

In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold is a generalization of a manifold...

.

The action of the group of homeomorphisms

Both Teichmüller space and the Riemann moduli space be more concisely defined in terms of a group action

Group action

In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...

. The set of all homeomorphisms underlies the group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

  whose binary operation

Binary operation

In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....

 is composition

Composition

Composition may refer to:* Composition , in which one assumes that a whole has a property solely because its various parts have that property* Compounding is also known as composition in linguistic literature* in computer science...

. The assignment is a group action

Group action

In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...

 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 . This subgroup acts on the set of complex structures, and the resulting orbit space is the Teichmüller space.

Relation to the mapping class group

The group is a normal subgroup

Normal subgroup

In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....

 of . The quotient group is called the mapping class group

Mapping class group

In mathematics, in the sub-field of geometric topology, the mapping class groupis an important algebraic invariant of a topological space. Briefly, the mapping class group is a discrete group of 'symmetries' of the space.-Motivation:...

 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 T_X

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 T_X 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 bewildering number of different natural metrics. These include:

Bergman metric

This is a special case of the Bergman metric

Bergman metric

In differential geometry, the Bergman metric is a Hermitian metric that can be defined on certain types of complex manifold. It is so called because it is derived from the Bergman kernel, both of which are named for Stefan Bergman.-Definition:...

 on any domain of holomorphy

Domain of holomorphy

In mathematics, in the theory of functions of several complex variables, a domain of holomorphy is a set which is maximal in the sense that there exists a holomorphic function on this set which cannot be extended to a bigger set....

.

Carathéodory metric

This is a special case of the Carathéodory metric

Carathéodory metric

In mathematics, the Carathéodory metric is a metric defined on the open unit ball of a complex Banach space that has many similar properties to the Poincaré metric of hyperbolic geometry. It is named after the Greek mathematician Constantin Carathéodory....

 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

Kobayashi metric

In mathematics, the original Kobayashi metric is a pseudometric on complex manifolds introduced by . It can be viewed as the dual of the Carathéodory metric, and has been extended to complex analytic spaces and almost complex manifolds...

 defined on any complex space. showed that it coincides with the Teichmüller metric.

McMullen metric

This is a complete Kähler metric of bounded sectional curvature introduced by that is Kähler-hyperbolic.

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 T_X, 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.

Thurston’s asymmetric metric

This is not a metric in the usual sense as it is not symmetric. It was introduced by .

Weil–Petersson metric

The Weil–Petersson metric

Weil–Petersson metric

In mathematics, the Weil–Petersson metric is a Kähler metric on the Teichmüller space Tg,n of genus g Riemann surfaces with n marked points...

 is a Riemannian metric on Teichmüller space. Ahlfors showed that it is a Kähler metric. It is not complete in general.

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 . 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

introduced a compactification whose points at infinity correspond to projective measured laminations. The compatified 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.

Examples of Teichmüller spaces

The Teichmüller spaces T_0,0, T_0,1, T_0,2, T_0,3 (corresponding to a sphere with at most 3 points removed) are points.

The Teichmüller spaces T_0,4, T_1,0, T_1,1, 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.