This talk is devoted to a series of papers we have published with T. Rivière. First, we have been interested in giving a unified proof of some "classical" quantization phenomena for problems such as harmonic maps, J-holomorphic curves or prescribed mean curvature. Then, since this proof relies only on the common dominator of theses problems, namely conformal invariance, we have been able to applied this theory to solve open questions in conformal geometry, such as Bi-harmonic maps, Willmore surfaces (with Y Bernard), free-boundary Harmonic maps (with R. Petrides).

We consider the motion of convex surfaces with normal speed given by arbitrary strictly monotone, homogeneous degree one functions of the principal curvatures (with no further smoothness assumptions). We prove that such processes deform arbitrary uniformly convex initial surfaces to points in finite time, with spherical limiting shape. This result was known previously only for smooth speeds and nonsmooth convex speeds. The crucial new ingredient in the argument, used to prove convergence of the rescaled surfaces to a sphere without requiring smoothness of the speed, is a surprising hidden divergence form structure in the evolution of certain curvature quantities.