Weil–Petersson class non-overlapping mappings into a Riemann surface

For a compact Riemann surface of genus g with n punctures, consider the class of n-tuples of conformal mappings (phi(1),..., phi(n)) of the unit disk each taking 0 to a puncture. Assume further that (1) these maps are quasiconformally extendible to C, (2) the pre-Schwarzian of each phi(i) is in the Bergman space, and (3) the images of the closures of the disk do not intersect. We show that the class of such non-overlapping mappings is a complex Hilbert manifold.