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

Abstract

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.