Weil-Petersson Teichmüller space revisited

Abstract

In our previous paper [23] it was proved that a sense-preserving homeomorphism g on the unit circle S1 belongs to the Weil-Petersson class WP(S1), namely, g can be extended to a quasiconformal mapping to the unit disk whose Beltrami coefficient is square integrable in the Poincaré metric if and only if g is absolutely continuous such that log⁡g′ belongs to the Sobolev class H12. In this sequel to [23], we show that the smooth Hilbert manifold structure on WP(S1) inherited from H12 by the pullback g↦log⁡|g′| is compatible with the standard Hilbert manifold structure introduced by Takhtajan-Teo [27]. This enables us to give a fast approach to some results in our previous papers [23] and [24].