Infinite dimensional geodesic flows and the universal Teichmüller space

The subject of this thesis lies in the intersection of differential geometry and functional analysis, a domain usually called global analysis. A central object in this work is the group Ds(M) of all orientation preserving diffeomorphisms of a compact manifold M with boundary. One of the main properties of this group is that it can be endowed with the structure of an infinite dimensional Hilbert manifold. The study of this object is motivated by the historical results obtained by Arnold [1966] and Ebin and Marsden [1970] in fluid mechanics. In Arnold [1966], it is shown that the motion of an incompressible fluid on a domain M can be formally described by a geodesic in the group Dμ(M) of all volume preserving diffeomorphisms of M, with respect to an L2 Riemannian metric associated to the kinetic energy of the fluid. The functional analytic study of this point of view was used in Ebin and Marsden [1970] in order to show that the Euler equations               u̇ + ∇uu = - grad p, div u = 0,   u || ∂M   are well-posed in Sobolev spaces with sufficient regularity, for any compact manifold M with boundary. We describe below the main results of this thesis. In Chapter 2 we show that the n-dimensional Camassa-Holm equations               ṁ + ∇um + ∇uTm + m div u = 0,   m = (1 - α2∆)u   are well-posed relative to Dirichlet, Navier-slip, or mixed boundary conditions. In order to obtain this result we use the remarkable fact that the associated Lagrangian motion describes a geodesic on a diffeomorphism group with respect to an H1 Riemannian metric. This allows us to use a method inspired from that of Ebin and Marsden [1970] for the Euler equations. The study of the analytic and geometric properties of Ds(M) is deeply related to results concerning the continuity of composition and multiplication in Sobolev spaces. In Chapter 3, we improve these results and obtain a theorem about the continuity of the multiplication and composition of functions below the critical exponent. This result will be used in a crucial way in Chapter 5. We study the classification of the coadjoint orbits of the Sobolev Bott-Virasoro group, for nonzero central charges. The Bott-Virasoro group is the unique nontrivial central extension of the group D(S1) of all diffeomorphisms of the circle. This group and its Lie algebra (the so called Virasoro algebra) appear in a natural way in different areas of physics and mathematics such as string theory or the Korteweg-de Vries equation. The classification of the Bott-Virasoro coadjoint orbits was carried out in Kirillov [1982], Witten [1988] et Balog, Fehér and Palla [1998] and others. In Chapter 4, we consider the completion of the Bott-Virasoro group with respect to a Sobolev topology and obtain a Hilbert manifold called the Sobolev Bott-Virasoro group. We determine the appropriate regularity assumption that allows an extension of the classification of the coadjoint orbits to the case of the Sobolev Bott-Virasoro, and show tha