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

Abstract. We investigate problems connected to the stability of the well-known Pohožaev obstruction. We generalize results which were obtained in the minimizing setting by Brezis and Nirenberg [2] and more recently in the radial situation by Brezis and Willem [3].Let be a smooth bounded domain in R n , n ≥ 3. Let h ∈ C 1 (R n ) and consider the equationIt is well-known that if is star-shaped with respect to the origin and if h satisfiesthen there are no non-trivial solutions of (0.1). This is a consequence of Pohožaev's identity (see [11] and equation (4.6) of appendix 4.3) and is referred to as the Pohožaev obstruction.The above equation has been quite intensively studied in the past thirty years. Many existence results have been obtained if is not assumed to be star-shaped or if h does not satisfy (0.2). It is almost impossible to give an exhaustive list of references on this equation.In this paper, we investigate the question of non-existence of positive solutions of equation (0.1) and more precisely the stability properties of the Pohožaev obstruction.Definition 0.1. Let be a star-shaped domain of R n and let (X, · X ) be some Banach space of functions on (typically X = C k,η ( ), X = L ∞ ( ) or X = L p ( )). Let h 0 ∈ X ∩C 1 ( ) be a function which satisfies (0.2). We say that the Pohožaev obstruction is X-stable at (h 0 , ) if the following property holds: there exists δ(h 0 , , X) > 0 such that for any function h ∈ X with h − h 0 X ≤ δ(h 0 , , X), the only non-negative C 2 -solution of (0.1) is u ≡ 0.O. Druet, P. Laurain: Département de mathématiques, UMPA,École normale supérieure de Lyon, 46 allée d'Italie,

Given a 3-dimensional Riemannian manifold (M, g), we prove that if (Φ k ) is a sequence of Willmore spheres (or more generally area-constrained Willmore spheres), having Willmore energy bounded above uniformly strictly by 8π, and Hausdorff converging to a pointp ∈ M , then Scal(p) = 0 and ∇ Scal(p) = 0 (resp. ∇ Scal(p) = 0). Moreover, a suitably rescaled sequence smoothly converges, up to subsequences and reparametrizations, to a round sphere in the euclidean 3-dimensional space. This generalizes previous results of Lamm and Metzger contained in [14]- [15]. An application to the Hawking mass is also established.

In this paper we prove a uniform estimate for the gradient of the Green function on a closed Riemann surface, independent of its conformal class, and we derive compactness results for immersions with L 2 -bounded second fundamental form and for riemannian surfaces of uniformly bounded gaussian curvature entropy.Math. Class. 32G15, 30F10, 53A05, 53A30, 35J35 1 4π in the sphere case and 1 in the torus case

We perform a replacement procedure in order to produce a free boundary minimal surface whose area achieves the min-max value over all disk sweepouts of a manifold whose boundary lies in a submanifold. Our result is based on a proof of the convexity of the energy for free boundary harmonic maps and a generalization of Colding-Minicozzi replacement procedure.