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

Francois Gay-Balmaz (1) , Tudor Ratiu (2)

1 LMD - Laboratoire de Météorologie Dynamique (UMR 8539)
2 EPFL - Ecole Polytechnique Fédérale de Lausanne

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

Domaines: Mathématiques [math]

Francois Gay-Balmaz, Tudor Ratiu. The geometry of the universal Teichmüller space and the Euler-Weil-Petersson equation. Advances in Mathematics, 2015, 279, pp.717-778. ⟨10.1016/j.aim.2015.04.005⟩. ⟨hal-01396618⟩