The Complex Structure of the Teichmuller Space

The Teichmuller 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 Teichmuller space. In this paper we describe the almost complex structure as a real matrix acting on the tangent space with basis (partial derivative/partial derivative L(A(1)), ... , partial derivative/partial derivative L, (A(n))). In the cotangent space the natural Hermitian scalar product of the associated quadratic differentials (Theta(A1), ... , Theta(An)) determines a skew-symmetric real matrix C and a symmetric matrix S. We prove that the matrix of the complex structure is SC-1.