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

Abstract

Given a continuous vector field $\lambda(t, \cdot)$ of Sobolev class $H^{\frac 32}$ on the unit circle $S^1$, the flow maps $\eta=g(t, \cdot)$ of the 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 continuously differentiable with respect to the Hilbert manifold structure of WP$(S^1)$ introduced by Takhtajan-Teo [TT]. The first assertion had already been demonstrated in our previous paper [Sh2]. In this sequel to [Sh2], we will continue to deal with the Weil-Petersson class WP$(S^1)$ and completely solve this conjecture in the affirmative.