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

TL;DR

This paper proves that flow curves generated by $H^{3/2}$-vector fields on the circle are in the Weil-Petersson class and are smoothly differentiable within that structure, confirming a recent conjecture.

Contribution

It fully resolves the conjecture that flow curves of $H^{3/2}$-vector fields are in WP$(S^1)$ and are smoothly differentiable in its Hilbert manifold structure.

Findings

Flow curves are in the Weil-Petersson class WP$(S^1)$.

Flow curves are continuously differentiable in WP$(S^1)$.

The conjecture by Gay-Balmaz-Ratiu is confirmed.

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.