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Given a continuous vector field $\lambda(t, \cdot)$ of Sobolev class$H^{\frac 32}$ on the unit circle $S^1$, the flow maps $\eta=g(t, \cdot)$ ofthe differential equation $$ \cases \frac{d\eta}{dt}=\lambda(t, \eta)\\\eta(0,\zeta)=\zeta \endcases $$ are known to be quasisymmetric homeomorphisms.Very recently, Gay-Balmaz-Ratiu [GR] conjectured that the flow curve $g(t,\cdot)$ is in the Weil-Petersson class WP$(S^1)$ and is continuouslydifferentiable with respect to the Hilbert manifold structure of WP$(S^1)$introduced by Takhtajan-Teo [TT]. The first assertion had already beendemonstrated in our previous paper [Sh2]. In this sequel to [Sh2], we willcontinue to deal with the Weil-Petersson class WP$(S^1)$ and completely solvethis conjecture in the affirmative.

Weil-Petersson Teichm\"{u}ller space II: smoothness of flow curves of $H^{\frac 32}$-vector fields