10/20/2006

Tristan Riviere
ETHZ, Zurich

Conservation laws for conformally invariant Lagrangian and  Schroedinger
systems

We will explain how to write 2-dimensional conformally invariant non-linear elliptic PDE (harmonic map equation, prescribed mean curvature equations...etc) in divergence form. These divergence-free quantities generalize to target manifolds without symmetries the well known conservation laws for weakly 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 enables us also to establish new results. In particular we solve a conjecture by E. Heinz asserting that the solutions to the prescribed bounded mean curvature equation in arbitrary manifolds are continuous. Our approach permits also to prove a more general conjecture by S.Hildebrandt claiming that critical points of continuously differentiable second order elliptic conformally invariant Lagrangian in two dimensions are continuous. We will explain how these results are deduced from a more general one on solutions to Schroedinger systems with antisymetric potentials.