The geometry of the universal Teichmuller space and the Euler-Weil-Petersson equation
On the identity component of the universal Teichmuller 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 H3/2-epsilon for all epsilon > 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. (C) 2015 Elsevier Inc. All rights reserved.
Record created on 2015-09-28, modified on 2016-08-09