The Complex Structure of the Teichmueller Space

摘要

The Teichmueller space of a topological surface X is a space that parameterizes complex structures on X up to the action of homeomorphisms that are isotopic to the identity. This space itself has a complex structure defined in terms of Beltrami differentials and quasi-conformal mappings. For X a surface of genus g and m punctures, n geodesics A_1,..., A_n (n = 6g - 6 + 2m) can be chosen so that their hyperbolic translation lengths (L(A_1),...,L(A_n)) give a local parameterization of the Teichmueller space. In this paper we describe the almost complex structure as a real matrix acting on the tangent space with basis (∂/∂L(A_1),..., ∂/∂L(A_n)). In the cotangent space the natural Hermitian scalar product of the associated quadratic differentials (Θ_(A_1),...,Θ_(A_n)) determines a skew-symmetric real matrix C and a symmetric matrix 5. We prove that the matrix of the complex structure is SC~(-1).