Conservation laws for conformal invariant variational problems

We succeed in writing 2-dimensional conformally invariant non-linear elliptic PDE (harmonic map equation, prescribed mean curvature equations...etc) in divergence form. This divergence free quantities generalize to target manifolds without symmetries the well known conservation laws for harmonic maps into homogeneous spaces. From this form we can recover, without the use of moving frame, all the classical regularity results known for 2-dimensional conformally invariant non-linear elliptic PDE . It enable us also to establish new results. In particular we solve a conjecture by E.Heinz asserting that the solutions to the precribed bounded mean curvature equation in arbitrary manifolds are continuous.Comment: 19 page

Entropy of semiclassical measures in dimension 2

We study the asymptotic properties of eigenfunctions of the Laplacian in the case of a compact Riemannian surface of Anosov type. We show that the Kolmogorov-Sinai entropy of a semiclassical measure for the geodesic flow is bounded from below by half of the Ruelle upper boundComment: 42 pages, 3 figures. Compared to the second version, I have removed the proof in the case of surfaces of nonpositive curvature and I have written it in a different article (arXiv:0911.1840

Perturbation of the semiclassical Schr\"odinger equation on negatively curved surfaces

We consider the semiclassical Schr\"odinger equation on a compact negatively curved surface. For any sequence of initial data microlocalized on the unit cotangent bundle, we look at the quantum evolution (below the Ehrenfest time) under small perturbations of the Schr\"odinger equation, and we prove that, in the semiclassical limit and for typical perturbations, the solutions become equidistributed on the unit cotangent bundle.Comment: 48 pages. Compared with version 1, we consider slightly different families of perturbations in order to simplify the expositio

On the string Lie algebra

We construct an abelian representative for the crossed module associated to the string Lie algebra. We show how to apply this construction in order to define quasi-invariant tensors which serve to categorify the infinitesimal braiding on the category of g-modules given by an r-matrix, following Cirio-Martins.Comment: 29 page