MODULI SPACES OF QUADRATIC-DIFFERENTIALS

The cotangent bundle of T(g, n) is a union of complex analytic subvarieties, V(pi), the level sets of the function "singularity pattern" of quadratic differentials. Each V(pi) is endowed with a natural affine complex structure and volume element. The latter contracts to a real analytic volume element, mu-pi, on the unit hypersurface, V1(pi), for the Teichmuller metric. mu-pi is invariant under the pure mapping class group, GAMMA-(g, n), and a certain class of functions is proved to be L(p)(mu-pi), 0 < p < 1, over the moduli space V1(pi)/GAMMA-(g, n). In particular, mu-pi(V1(pi)/GAMMA-(g, n)) < infinity, a statement which generalizes a theorem by H. Masur.