Teichmüller space
Definition
Teichmüller space, named after the German mathematician Oswald Teichmüller, is a mathematical concept in the field of complex analysis and differential geometry. It is a parameter space that is associated with the structures of Riemann surfaces or, more generally, complex structures on a given topological surface.
Mathematical Background
The concept of Teichmüller space is rooted in the study of moduli spaces, which are geometric spaces whose points represent equivalence classes of certain types of mathematical objects. In the case of Teichmüller space, the objects in question are complex structures on a given topological surface.
Construction
The construction of Teichmüller space involves several steps. First, one must define a complex structure on a topological surface. This involves choosing a homeomorphism class of complex structures, which is a set of structures that can be continuously deformed into one another.
The next step is to define a quasiconformal mapping, which is a homeomorphism between two Riemann surfaces that is 'almost' conformal, meaning it preserves angles to first order. The set of all quasiconformal mappings between two given Riemann surfaces forms a group under composition, known as the Teichmüller modular group.
Finally, the Teichmüller space is defined as the set of equivalence classes of marked Riemann surfaces, where two marked surfaces are considered equivalent if there is a quasiconformal mapping between them that commutes with the marking.
Properties
Teichmüller space has many interesting properties. It is a complex manifold, meaning it locally resembles complex Euclidean space. Its dimension is equal to the number of independent complex parameters needed to specify a point in the space, which is equal to the genus of the underlying topological surface.
Teichmüller space also has a natural metric, known as the Teichmüller metric, which is defined in terms of the extremal length of curves on the surface. This metric makes Teichmüller space into a metric space, and it is in fact a complete metric space, meaning that every Cauchy sequence of points in the space converges to a limit in the space.
Applications
Teichmüller space has found applications in several areas of mathematics. In complex analysis, it is used to study the behavior of holomorphic functions on Riemann surfaces. In differential geometry, it is used to study the geometry of surfaces and higher-dimensional manifolds. It also plays a role in string theory, where it is used to describe the space of possible shapes of a string's worldsheet.