Teichmüller room

This article deals with the module space of Riemann surfaces, for the module space of hyperbolic metrics see Fricke space .

In function theory , the Teichmüller space (according to Oswald Teichmüller ) designates a space of equivalence classes of compact Riemann surfaces and thus enables a classification of all compact Riemann surfaces.

definition

Let be a compact Riemann surface with gender and conformal structure . Two structures on the same face are said to be equivalent if there is a conformal diffeomorphism that is homotopic to identity . The space of all these equivalence classes from Riemann surfaces to gender is called the Teichmüller space and is denoted by. ${\displaystyle (X,g)}$ ${\displaystyle p\in \mathbb {N} _{0}}$ ${\displaystyle g}$ ${\displaystyle (X,g_{1}),(X,g_{2})}$ ${\displaystyle f\colon X\to X}$ ${\displaystyle p\in \mathbb {N} _{0}}$ ${\displaystyle {\mathcal {T}}_{p}(X)}$

classification

After a set of Teichmuller is provided with a mating structure of a manifold, for each compliant structure diffeomorphic to finite-dimensional vector space of the quadratic differential forms on whose dimension is calculated as follows: ${\displaystyle {\mathcal {T}}_{p}(X)}$ ${\displaystyle g}$ ${\displaystyle Q_{g}(X)}$ ${\displaystyle X}$

• ${\displaystyle \mathrm {dim} _{\mathbb {C} }\;Q_{g}(X)=0}$if ${\displaystyle p=0}$
• ${\displaystyle \mathrm {dim} _{\mathbb {C} }\;Q_{g}(X)=1}$if ${\displaystyle p=1}$
• ${\displaystyle \mathrm {dim} _{\mathbb {C} }\;Q_{g}(X)=3p-3}$if ${\displaystyle p\geq 2}$

A mapping between Riemann surfaces is holomorphic if and only if it is conformal (true to angle) and orientation preserving. Thus, the classification of the complex structures can also be obtained from the classification of the conformal structures.

motivation

• For a compact Riemann surface of gender, there is a natural bijective relationship between the conformal structures and the hyperbolic metrics that can be defined on that surface. Thus, the problem of possible conformal structures can be traced back to a geometrical-analytical question of metrics. The hyperbolic metrics are induced by the universal overlay by the hyperbolic half-plane . ${\displaystyle p\geq 2}$ ${\displaystyle \mathbb {H} }$
• The space of all equivalence classes of possible conformal structures on a gender surface has a complicated topology and is not a manifold; where two structures are considered equivalent if a conformal mapping exists between them. This motivates the weaker equivalence relation of the Teichmüller area. ${\displaystyle {\mathcal {M}}_{p}(X)}$ ${\displaystyle X}$ ${\displaystyle p}$ ${\displaystyle (X,g_{1}),(X,g_{2})}$
• There is for each compliant structure a bijective mapping in the space of the quadratic differential forms on which apparently forms a vector space and, moreover, is finite-dimensional. This ultimately defines a differentiability structure and is diffeomorphic to a finite-dimensional vector space. This last step is essentially Teichmüller's sentence formulated above. ${\displaystyle g}$ ${\displaystyle q_{g}:{\mathcal {T}}_{p}(X)\to Q_{g}(X)}$ ${\displaystyle Q_{g}(X)}$ ${\displaystyle (X,g)}$ ${\displaystyle {\mathcal {T}}_{p}(X)}$ ${\displaystyle {\mathcal {T}}_{p}(X)}$

Higher Teichmüller theory

The holonomy representation embeds the Teichmüller space in the quotient ${\displaystyle {\mathcal {T}}_{p}(X)}$

${\displaystyle \operatorname {Hom} (\pi _{1}X,\operatorname {PSL} (2,\mathbb {R} ))/\operatorname {PSL} (2,\mathbb {R} )}$

the Darstellungsvarietät , wherein an on works by conjugation. This embedding identifies the Teichmüller space with the number of injective, discrete representations. The latter form a coherent component of the representation variety and can also be characterized by various other conditions. The term Higher Teichmüller Theory summarizes approaches with which individual components of the representation variety are to be characterized for higher-dimensional Lie groups and compact surfaces X. ${\displaystyle \operatorname {PSL} (2,\mathbb {R} )}$ ${\displaystyle \operatorname {Hom} (\pi _{1}X,\operatorname {PSL} (2,\mathbb {R} ))}$ ${\displaystyle G}$ ${\displaystyle \operatorname {Hom} (\pi _{1}X,G)/G}$

literature

• Jürgen Jost: Compact Riemann Surfaces . Springer Verlag, 2006, ISBN 3-540-33065-8
• Athanase Papadopoulos (Ed.): Handbook of Teichmüller theory. Vol. I, European Mathematical Society (EMS), Zurich 2007, ISBN 978-3-03719-029-6 , doi: 10.4171 / 029 . (IRMA Lectures in Mathematics and Theoretical Physics 11)
• Athanase Papadopoulos (Ed.): Handbook of Teichmüller theory. Vol. II, European Mathematical Society (EMS), Zurich 2009, ISBN 978-3-03719-055-5 , doi: 10.4171 / 055 . (IRMA Lectures in Mathematics and Theoretical Physics 13)
• Athanase Papadopoulos (Ed.): Handbook of Teichmüller theory. Vol. III, European Mathematical Society (EMS), Zurich 2012, ISBN 978-3-03719-103-3 , doi: 10.4171 / 103 . (IRMA Lectures in Mathematics and Theoretical Physics 19)

Web links

• Kapovich: Notes on Teichmuller Theory . (PDF)

Individual evidence

1. ^ Burger, Iozzi, Wienhard: Higher Teichmüller spaces: From SL (2, R) to other Lie groups . In: Handbook of Teichmüller theory . Vol. III, arxiv : 1004.2894v4