In the present work we establish a quantization result for the angular part of the energy of solutions to elliptic linear systems of Schrödinger type with antisymmetric potentials in two dimension. This quantization is a consequence of uniform Lorentz-Wente type estimates in degenerating annuli. We derive from this angular quantization the full energy quantization for general critical points to functionals which are conformally invariant or also for pseudo-holomorphic curves on degenerating Riemann surfaces.A.M.S. Classification :

In the present work we establish an energy quantization (or energy identity) result for solutions to scaling invariant variational problems in dimension 4 which includes biharmonic maps (extrinsic and intrinsic). To that end we first establish an angular energy quantization for solutions to critical linear 4th order elliptic systems with antisymmetric potentials. The method is inspired by the one introduced by the authors previously in "Angular energy quantization for linear elliptic systems with antisymmetric potentials and applications" (2011) for 2nd order problems.

We establish an energy quantization result for sequences of Willmore surfaces when the underlying sequence of Riemann surfaces is degenerating in the moduli space. We notably exhibit a new residue which quantifies the potential loss of energy in collar regions. Thanks to these residues, we also establish the compactness (modulo the action of the Möbius group of conformal transformations of R 3 ∪ {∞}) of the space of Willmore immersions of any arbitrary closed 2-dimensional oriented manifold into R 3 with uniformly bounded conformal class and energy below 12π.

We prove a quantification result for harmonic maps with free boundary from arbitrary Riemannian surfaces into the unit ball of R n+1 with bounded energy. This generalizes results obtained by Da Lio [1] on the disc.Let (M, g) be a smooth Riemannian surface with a smooth nonempty boundary with s connected components. We fix n ≥ 2 and let B n+1 be the unit ball of R n+1 . A map u : (M, g) → B n+1 is a smooth harmonic map with free boundary if it is harmonic and smooth up to the boundary, u(∂M ) ⊂ S n and ∂ ν u is parallel to u, (or ∂ ν u ∈ (T u S n ) ⊥ ). The energy of such a map is defined as