Invariant Metrics on Teichmüller Space

Abstract

This chapter discusses the differential metrics and Hermitian metrics on Teichmüller space. From the Ahlfors–Bers–Teichmüller theory, it is known that Tg with the complex structure it inherits from the space of Beltrami differentials is a (3g–3)-dimensional complex manifold, which is equivalent to a contractible bounded domain in ℂ3g–3. The tangent space to Tg at a point W of Tg consists of all Beltrami differentials on W modulo, those which are infinitesimally trivial. As μ is infinitesimally trivial, if and only if ∫wμQ, = 0 for each holomorphic quadratic differential Q on W, the cotangent space to Tg at W is given by the space Q(W) of all quadratic differentials on W. The norm on the tangent space will be Hermitian if and only if the norm on Q(W) is, and a pseudonorm will be Hermitian if and only if the norm on the distinguished subspace S ⊂ Q is Hermitian. Thus, the Hermitian pseudonorms are obtained by specifying an inner product on a subspace S ⊂ Q(W). Such a pseudometric will be invariant on Tg if the norm depends only on the conformai structure of W.