On the Kähler form of the moduli space of once punctured tori

Abstract

A Riemann surface R of negative Euler characteristic has a unique hyperbolic metric. Provided R has finite area in this metric the Teichmiiller space T(R) of R will be a complex manifold. The complex structure of T(R) is characterized by describing the holomorphic cotangent space at R. A natural identification exists of the holomorphic cotangent space and Q(R), the space of holomorphic quadratic differentials on R. Consequently a Hermitian structure on Q(R) naturally gives rise to one on T(R). An example is the Petersson inner product. Given ~0, ~b ~ Q(R) define