In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Teichmüller space
of a (real) topological (or differential)
surface is a space that parametrizes
complex structures on
up to the action of
homeomorphism
In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...
s that are
isotopic to the
identity homeomorphism. Teichmüller spaces are named after
Oswald Teichmüller.
Each point in a Teichmüller space
may be regarded as an isomorphism class of "marked"
Riemann surfaces, where a "marking" is an isotopy class of homeomorphisms from
to itself. It can be viewed as a
moduli space for marked
hyperbolic structure on the surface, and this endows it with a natural topology for which it is homeomorphic to a
ball
A ball is a round object (usually spherical, but sometimes ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used for s ...
of dimension
for a surface of
genus
Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...
. In this way Teichmüller space can be viewed as the
universal covering orbifold of the
Riemann moduli space.
The Teichmüller space has a canonical
complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such th ...
structure and a wealth of natural
metrics. The study of geometric features of these various structures is an active body of research.
The sub-field of mathematics that studies the Teichmüller space is called Teichmüller theory.
History
Moduli spaces for
Riemann surfaces and related
Fuchsian groups have been studied since the work of
Bernhard Riemann
Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the f ...
(1826–1866), who knew that
parameters were needed to describe the variations of complex structures on a surface of genus
. The early study of Teichmüller space, in the late nineteenth–early twentieth century, was geometric and founded on the interpretation of Riemann surfaces as hyperbolic surfaces. Among the main contributors were
Felix Klein
Felix Christian Klein (; ; 25 April 1849 – 22 June 1925) was a German mathematician and Mathematics education, mathematics educator, known for his work in group theory, complex analysis, non-Euclidean geometry, and the associations betwe ...
,
Henri Poincaré
Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philosophy of science, philosopher of science. He is often described as a polymath, and in mathemati ...
,
Paul Koebe,
Jakob Nielsen,
Robert Fricke and
Werner Fenchel.
The main contribution of Teichmüller to the study of moduli was the introduction of
quasiconformal mappings to the subject. They allow us to give much more depth to the study of moduli spaces by endowing them with additional features that were not present in the previous, more elementary works. After World War II the subject was developed further in this analytic vein, in particular by
Lars Ahlfors and
Lipman Bers. The theory continues to be active, with numerous studies of the complex structure of Teichmüller space (introduced by Bers).
The geometric vein in the study of Teichmüller space was revived following the work of
William Thurston
William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds.
Thurst ...
in the late 1970s, who introduced a geometric compactification which he used in his study of the
mapping class group
In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space.
Mo ...
of a surface. Other more combinatorial objects associated to this group (in particular the
curve complex) have also been related to Teichmüller space, and this is a very active subject of research in
geometric group theory.
Definitions
Teichmüller space from complex structures
Let
be an
orientable smooth surface (a
differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ...
of dimension 2). Informally the Teichmüller space
of
is the space of
Riemann surface structures on
up to
isotopy.
Formally it can be defined as follows. Two
complex structures on
are said to be equivalent if there is a
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable.
Definit ...
such that:
* It is holomorphic (the differential is complex linear at each point, for the structures
at the source and
at the target) ;
* it is isotopic to the identity of
(there is a continuous map
such that
).
Then
is the space of equivalence classes of complex structures on
for this relation.
Another equivalent definition is as follows:
is the space of pairs
where
is a Riemann surface and
a diffeomorphism, and two pairs
are regarded as equivalent if
is isotopic to a holomorphic diffeomorphism. Such a pair is called a ''marked Riemann surface''; the ''marking'' being the diffeomorphism; another definition of markings is by systems of curves.
There are two simple examples that are immediately computed from the
Uniformization theorem
In mathematics, the uniformization theorem states that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. The theorem is a generali ...
: there is a unique complex structure on the
sphere
A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
(see
Riemann sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann,
is a Mathematical model, model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents ...
) and there are two on
(the complex plane and the
unit disk
In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1:
:D_1(P) = \.\,
The closed unit disk around ''P'' is the set of points whose d ...
) and in each case the group of positive diffeomorphisms is connected. Thus the Teichmüller space of
is a single point and that of
contains exactly two points.
A slightly more involved example is the open
annulus, for which the Teichmüller space is the interval
(the complex structure associated to
is the Riemann surface
).
The Teichmüller space of the torus and flat metrics
The next example is the torus
In this case any complex structure can be realised by a Riemann surface of the form
(a complex Elliptic curve#Elliptic curves over the complex numbers, elliptic curve) for a
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
where
:
is the complex upper half-plane. Then we have a
bijection
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
:
:
:
and thus the Teichmüller space of
is
If we identify
with the
Euclidean plane
In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...
then each point in Teichmüller space can also be viewed as a marked
flat structure on
Thus the Teichmüller space is in bijection with the set of pairs
where
is a flat surface and
is a diffeomorphism up to isotopy on
.
Finite type surfaces
These are the surfaces for which Teichmüller space is most often studied, which include closed surfaces. A surface is of finite type if it is diffeomorphic to a compact surface minus a finite set. If
is a
closed surface of
genus
Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...
then the surface obtained by removing
points from
is usually denoted
and its Teichmüller space by
Teichmüller spaces and hyperbolic metrics
Every finite type orientable surface other than the ones above admits
complete Riemannian metric
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
s of constant curvature
. For a given surface of finite type there is a bijection between such metrics and complex structures as follows from the
uniformisation theorem. Thus if
the Teichmüller space
can be realised as the set of marked
hyperbolic surfaces of genus
with
cusps, that is the set of pairs
where
is a hyperbolic surface and
is a diffeomorphism, modulo the equivalence relation where
and
are identified if
is isotopic to an isometry.
The topology on Teichmüller space
In all cases computed above there is an obvious topology on Teichmüller space. In the general case there are many natural ways to topologise
, perhaps the simplest is via hyperbolic metrics and length functions.
If
is a
closed curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
on
and
a marked hyperbolic surface then one
is homotopic to a unique
closed geodesic on
(up to parametrisation). The value at
of the ''length function'' associated to (the homotopy class of)
is then:
:
Let
be the set of
simple closed curves on
. Then the map
:
:
is an embedding. The space
has the
product topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemin ...
and
is endowed with the
induced topology. With this topology
is homeomorphic to
In fact one can obtain an embedding with
curves, and even
. In both case one can use the embedding to give a geometric proof of the homeomorphism above.
More examples of small Teichmüller spaces
There is a unique complete hyperbolic metric of finite volume on the three-holed sphere and so the Teichmüller space of finite-volume complete metrics of constant curvature
is a point (this also follows from the dimension formula of the previous paragraph).
The Teichmüller spaces
and
are naturally realised as the upper half-plane, as can be seen using Fenchel–Nielsen coordinates.
Teichmüller space and conformal structures
Instead of complex structures or hyperbolic metrics one can define Teichmüller space using
conformal structures. Indeed, conformal structures are the same as complex structures in two (real) dimensions. Moreover, the Uniformisation Theorem also implies that in each conformal class of Riemannian metrics on a surface there is a unique metric of
constant curvature.
Teichmüller spaces as representation spaces
Yet another interpretation of Teichmüller space is as a representation space for surface groups. If
is hyperbolic, of finite type and
is the
fundamental group of
then Teichmüller space is in natural bijection with:
*The set of injective representations
with discrete image, up to conjugation by an element of
, if
is compact ;
*In general, the set of such representations, with the added condition that those elements of
which are represented by curves freely homotopic to a puncture are sent to
parabolic elements of
, again up to conjugation by an element of
.
The map sends a marked hyperbolic structure
to the composition
where
is the
monodromy of the hyperbolic structure and
is the isomorphism induced by
.
Note that this realises
as a closed subset of
which endows it with a topology. This can be used to see the homeomorphism
directly.
This interpretation of Teichmüller space is generalised by
higher Teichmüller theory, where the group
is replaced by an arbitrary semisimple
Lie group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.
A manifold is a space that locally resembles Eucli ...
.
A remark on categories
All definitions above can be made in the
topological category instead of the category of differentiable manifolds, and this does not change the objects.
Infinite-dimensional Teichmüller spaces
Surfaces which are not of finite type also admit hyperbolic structures, which can be parametrised by infinite-dimensional spaces (homeomorphic to
). Another example of infinite-dimensional space related to Teichmüller theory is the Teichmüller space of a lamination by surfaces.
Action of the mapping class group and relation to moduli space
The map to moduli space
There is a map from Teichmüller space to the
moduli space of Riemann surfaces diffeomorphic to
, defined by
. It is a covering map, and since
is
simply connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every Path (topology), path between two points can be continuously transformed into any other such path while preserving ...
it is the
orbifold universal cover for the moduli space.
Action of the mapping class group
The
mapping class group
In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space.
Mo ...
of
is the coset group
of the
diffeomorphism group of
by the normal subgroup of those that are isotopic to the identity (the same definition can be made with homeomorphisms instead of diffeomorphisms and, for surfaces, this does not change the resulting group). The group of diffeomorphisms acts naturally on Teichmüller space by
:
If
is a mapping class and
two diffeomorphisms representing it then they are isotopic. Thus the classes of
and
are the same in Teichmüller space, and the action above factorises through the mapping class group.
The action of the mapping class group
on the Teichmüller space is
properly discontinuous, and the quotient is the moduli space.
Fixed points
The Nielsen realisation problem asks whether any finite subgroup of the mapping class group has a global fixed point (a point fixed by all group elements) in Teichmüller space. In more classical terms the question is: can every finite subgroup of
be realised as a group of isometries of some complete hyperbolic metric on
(or equivalently as a group of holomorphic diffeomorphisms of some complex structure). This was solved by
Steven Kerckhoff.
Coordinates
Fenchel–Nielsen coordinates
The Fenchel–Nielsen coordinates (so named after
Werner Fenchel and
Jakob Nielsen) on the Teichmüller space
are associated to a
pants decomposition of the surface
. This is a decomposition of
into
pairs of pants, and to each curve in the decomposition is associated its length in the hyperbolic metric corresponding to the point in Teichmüller space, and another real parameter called the twist which is more involved to define.
In case of a closed surface of genus
there are
curves in a pants decomposition and we get
parameters, which is the dimension of
. The Fenchel–Nielsen coordinates in fact define a homeomorphism
.
In the case of a surface with punctures some pairs of pants are "degenerate" (they have a cusp) and give only two length and twist parameters. Again in this case the Fenchel–Nielsen coordinates define a homeomorphism
.
Shear coordinates
If
the surface
admits ideal Triangulation (geometry), triangulations (whose vertices are exactly the punctures). By the formula for the
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's ...
such a triangulation has
triangles. A hyperbolic structure
on
determines a (unique up to isotopy) diffeomorphism
sending every triangle to a hyperbolic
ideal triangle, thus a point in
. The parameters for such a structure are the translation lengths for each pair of sides of the triangles glued in the triangulation. There are
such parameters which can each take any value in
, and the completeness of the structure corresponds to a linear equation and thus we get the right dimension
. These coordinates are called ''shear coordinates''.
For closed surfaces, a pair of pants can be decomposed as the union of two ideal triangles (it can be seen as an incomplete hyperbolic metric on the three-holed sphere). Thus we also get
shear coordinates on
.
Earthquakes
A simple ''earthquake path'' in Teichmüller space is a path determined by varying a single shear or length Fenchel–Nielsen coordinate (for a fixed ideal triangulation of a surface). The name comes from seeing the ideal triangles or the pants as
tectonic plates and the shear as plate motion.
More generally one can do earthquakes along geodesic
laminations. A theorem of Thurston then states that two points in Teichmüller space are joined by a unique earthquake path.
Analytic theory
Quasiconformal mappings
A quasiconformal mapping between two Riemann surfaces is a homeomorphism which deforms the conformal structure in a bounded manner over the surface. More precisely it is differentiable almost everywhere and there is a constant
, called the ''dilatation'', such that
:
where
are the derivatives in a conformal coordinate
and its conjugate
.
There are quasi-conformal mappings in every isotopy class and so an alternative definition for The Teichmüller space is as follows. Fix a Riemann surface
diffeomorphic to
, and Teichmüller space is in natural bijection with the marked surfaces
where
is a quasiconformal mapping, up to the same equivalence relation as above.
Quadratic differentials and the Bers embedding

With the definition above, if
there is a natural map from Teichmüller space to the space of
-equivariant solutions to the Beltrami differential equation. These give rise, via the Schwarzian derivative, to
quadratic differentials on
. The space of those is a complex space of complex dimension
, and the image of Teichmüller space is an open set. This map is called the Bers embedding.
A quadratic differential on
can be represented by a
translation surface conformal to
.
Teichmüller mappings
Teichmüller's theorem states that between two marked Riemann surfaces
and
there is always a unique quasiconformal mapping
in the isotopy class of
which has minimal dilatation. This map is called a Teichmüller mapping.
In the geometric picture this means that for every two diffeomorphic Riemann surfaces
and diffeomorphism
there exists two polygons representing
and an affine map sending one to the other, which has smallest dilatation among all quasiconformal maps
.
Metrics
The Teichmüller metric
If
and the Teichmüller mapping between them has dilatation
then the Teichmüller distance between them is by definition
. This indeed defines a distance on
which induces its topology, and for which it is complete. This is the metric most commonly used for the study of the metric geometry of Teichmüller space. In particular it is of interest to geometric group theorists.
There is a function similarly defined, using the
Lipschitz constants of maps between hyperbolic surfaces instead of the quasiconformal dilatations, on
, which is not symmetric.
The Weil–Petersson metric
Quadratic differentials on a Riemann surface
are identified with the cotangent space at
to Teichmüller space. The Weil–Petersson metric is the Riemannian metric defined by the
inner product on quadratic differentials.
Compactifications
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.
William Thurston
William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds.
Thurst ...
later found a compactification without this disadvantage, which has become the most widely used 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, 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.
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.
Gardiner–Masur compactification
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.
Large-scale geometry
There has been an extensive study of the geometric properties of Teichmüller space endowed with the Teichmüller metric. Known large-scale properties include:
*Teichmüller space
contains flat subspaces of dimension
, and there are no higher-dimensional quasi-isometrically embedded flats.
*In particular, if
or
or
then
is not
hyperbolic.
On the other hand, Teichmüller space exhibits several properties characteristic of hyperbolic spaces, such as:
*Some geodesics behave like they do in hyperbolic space.
*Random walks on Teichmüller space converge almost surely to a point on the Thurston boundary.
Some of these features can be explained by the study of maps from Teichmüller space to the curve complex, which is known to be hyperbolic.
Complex geometry
The Bers embedding gives
a complex structure as an open subset of
Metrics coming from the complex structure
Since Teichmüller space is a complex manifold it carries a
Carathéodory metric. Teichmüller space is Kobayashi hyperbolic and its
Kobayashi metric coincides with the Teichmüller metric. This latter result is used in Royden's proof that the mapping class group is the full group of isometries for the Teichmüller metric.
The Bers embedding realises Teichmüller space as a
domain of holomorphy
In mathematics, in the theory of functions of several complex variables, a domain of holomorphy is a domain which is maximal in the sense that there exists a holomorphic function on this domain which cannot be extended to a bigger domain.
Forma ...
and hence it also carries a
Bergman metric.
Kähler metrics on Teichmüller space
The Weil–Petersson metric is Kähler but it is not complete.
Cheng and
Yau showed that there is a unique complete
Kähler–Einstein metric on Teichmüller space. It has constant negative scalar curvature.
Teichmüller space also carries a complete Kähler metric of bounded sectional curvature introduced by that is Kähler-hyperbolic.
Equivalence of metrics
With the exception of the incomplete Weil–Petersson metric, all metrics on Teichmüller space introduced here are
quasi-isometric to each other.
See also
*
Moduli of algebraic curves
*
p-adic Teichmüller theory
*
Inter-universal Teichmüller theory
*
Teichmüller modular form
References
Sources
*
*
*
*
*
*
*
*
*
Further reading
*
*
*
* The last volume contains translations of several of Teichmüller's papers.
*
*
*
{{DEFAULTSORT:Teichmuller Space
Riemann surfaces
Moduli theory
Differential geometry
Geometric group theory