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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 H^(3/2−ε) for all ε > 0. The evolutionary PDE associ-ated 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
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Francois Gay-Balmaz, Tudor Ratiu. The geometry of the universal Teichmüller space and the Euler-Weil-Petersson equation. Advances in Mathematics, Elsevier, 2015, 279, pp.717-778. ⟨10.1016/j.aim.2015.04.005⟩. ⟨hal-01396618⟩

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