Abstract
Given a continuous vector field λ(t,⋅) of Sobolev class H32 on the unit circle S1, the flow maps η=g(t,⋅) of the differential equation{dηdt=λ(t,η)η(0,ζ)=ζ are known to be quasisymmetric homeomorphisms. Very recently, Gay-Balmaz-Ratiu [15] conjectured that the flow curve g(t,⋅) is in the Weil-Petersson class WP(S1) and is continuously differentiable with respect to the Hilbert manifold structure of WP(S1) introduced by Takhtajan-Teo [40]. The first assertion had already been demonstrated in our previous paper [36]. In this sequel to [36], we will continue to deal with the Weil-Petersson class WP(S1) and completely solve this conjecture in the affirmative.
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