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

Uniqueness of tangent cones for calibrated 2-cycles

We prove that tangent cones to 2-dimensional calibrated cycles are unique. Using this result we prove a rate of convergence for the mass of the blow-up of a calibrated integral 2-cycle towards the limiting density. With the same techniques, we can also prove such a rate for J-holomorphic maps between almost complex manifolds and deduce the uniqueness of their tangent maps.Comment: 37 page

Vortex energy and vortex bending for a rotating Bose-Einstein condensate

For a Bose-Einstein condensate placed in a rotating trap, we give a simplified expression of the Gross-Pitaevskii energy in the Thomas Fermi regime, which only depends on the number and shape of the vortex lines. Then we check numerically that when there is one vortex line, our simplified expression leads to solutions with a bent vortex for a range of rotationnal velocities and trap parameters which are consistent with the experiments.Comment: 7 pages, 2 figures. submitte