Geometric Aspects of the Moduli Space of Riemann Surfaces
نویسندگان
چکیده
The study of moduli space and Teichmüller space has a long history. These two spaces lie in the intersections of the researches in many areas of mathematics and physics. Many deep results have been obtained in history by many famous mathematicians. Here we will only mention a few that are closely related to our discussions. Riemann was the first who considered the space M of all complex structures on an orientable surface modulo the action of orientation preserving diffeomorphisms. He derived the dimension of this space dimR M = 6g − 6 where g ≥ 2 is the genus of the topological surface. In 1940’s, Teichmüller considered a cover ofM by taking the quotient of all complex structures by those orientation preserving diffeomorphims which are isotopic to the identity map. The Teichmüller space Tg is a contractible set in C3g−3. Furthermore, it is a pseudoconvex domain. Teichmüller also introduced the Teichmüller metric by first taking the L1 norm on the cotangent space of Tg and then taking the dual norm on the tangent space. This is a Finsler metric. Two other interesting Finsler metrics are the Carathéodory metric and the Kobayashi metric. These Finsler metrics have been powerful tools to study the hyperbolic property of the moduli and the Teichmüller spaces and the mapping class groups. For example in 1970’s Royden proved that the Teichmüller metric and the Kobayashi metric are the same, and as a corollary he proved the famous result that the holomorphic automorphism group of the Teichmüller space is exactly the mapping class group.
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