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The primary purpose of the paper is to study how a Riemann mapping depends onthe corresponding Jordan curve. We are mainly concerned with those Jordancurves in the Weil-Petersson class, namely, the corresponding Riemann mappingscan be quasiconformally extended to the whole plane with Beltrami coefficientsbeing square integrable under the Poincar\'e metric. We endow the space of allnormalized Weil-Petersson curves with a new real Hilbert manifold structure andshow that it is topologically equivalent to the standard complex Hilbertmanifold structure.

Weil-Petersson Teichm\"{u}ller space III: dependence of Riemann mappings for Weil-Petersson curves