The geometry of the universal Teichmüller space and the Euler-Weil-Petersson equation

On the identity component of the universal Teichmüller space endowed with the Takhtajan-Teo topology, the geodesics of the Weil-Petersson metric are shown to exist for all time. This component is naturally a subgroup of the quasisymmetric homeomorphisms of the circle. Viewed this way, the regularity of its elements is shown to be H32-ε for all ε. >. 0. The evolutionary PDE associated to the spatial representation of the geodesics of the Weil-Petersson metric is derived using multiplication and composition below the critical Sobolev index 3/2. Geodesic completeness is used to introduce special classes of solutions of this PDE analogous to peakons. Our setting is used to prove that there exists a unique geodesic between each two shapes in the plane in the context of the application of the Weil-Petersson metric in imaging.