  The primary purpose of the paper is to study how a Riemann mapping depends on
the corresponding Jordan curve. We are mainly concerned with those Jordan
curves in the Weil-Petersson class, namely, the corresponding Riemann mappings
can be quasiconformally extended to the whole plane with Beltrami coefficients
being square integrable under the Poincar\'e metric. We endow the space of all
normalized Weil-Petersson curves with a new real Hilbert manifold structure and
show that it is topologically equivalent to the standard complex Hilbert
manifold structure.
