Theorem II.1 [Mor1] Let u be a map in the Sobolev space C 0 ∩ W 1,2 (D 2 , R m ) and let ε > 0, then there exists an homeomorphism Ψ of the disc such that Ψ ∈ W 1,2 (D 2 , D 2 ),

: We found a new formulation to the Euler-Lagrange equation of the Willmore functional for immersed surfaces in R m . This new formulation of Willmore equation appears to be of divergence form, moreover, the nonlinearities are made of jacobians. Additionally to that, if H denotes the mean curvature vector of the surface, this new form writes L H = 0 where L is a well defined locally invertible self-adjoint operator. These 3 facts have numerous consequences in the analysis of Willmore surfaces. One first consequence is that the long standing open problem to give a meaning to the Willmore EulerLagrange equation for immersions having only L 2 bounded second fundamental form is now solved. We then establish the regularity of weak W 2,p −Willmore surfaces for any p for which the Gauss map is continuous : p > 2. This is based on the proof of an ǫ−regularity result for weak Willmore surfaces. We establish then a weak compactness result for Willmore surfaces of energy less than 8π − δ for every δ > 0. This theorem is based on a point removability result we prove for Wilmore surfaces in R m . This result extends to arbitrary codimension the main result in [KS3] established for surfaces in R 3 . Finally, we deduce from this point removability result the strong compactness, modulo the Möbius group action, of Willmore tori below the energy level 8π − δ in dimensions 3 and 4. The dimension 3 case was already solved in [KS3]. I IntroductionWeak formulations of PDE offer not only the possibility to enlarge the class of solutions to the space of singular solutions but also provide a flexible setting in which the analysis of smooth solutions becomes much more efficient. This is the idea that we want to illustrate in this paper by introducing this new weak formulation of Willmore surfaces.For a given oriented surface Σ and a smooth positive immersion Φ of Σ into the Euclidian space R m , for some m ≥ 3, we introduce first the Gauss map n from Σ into Gr m−2 (R m ), the grassmanian of oriented m − 2−planes of R m , which to every point x in Σ assigns the unit m − 2-unit vector defining the m − 2−plane N Φ(x) Φ(Σ) orthogonal to the oriented tangent space T Φ(x) Φ(Σ). This map n from Σ into Gr m−2 (R m ) defines a projection map π n : for everyLet then B x be the second fundamental form of the immersion Φ of Σ. B x is a symmetric bilinear form on T x Σ with values into N Φ(x) Φ(Σ). B x is given * Department of Mathematics, ETH Zentrum, CH-8093 Zürich, Switzerland.By the mean of the ambiant scalar product in R m , which induces a metric g on Σ, we define the trace of B x , tr( B x ), which is a vector in N Φ(x) Φ(Σ) given by tr( B x ) = B x (e 1 , e 1 ) + B x (e 2 , e 2 ) where (e 1 , e 2 ) is an arbitrary orthonormal basis of T x Σ. The mean curvature vector H(x) at x of the immersion by Φ of Σ is with theses notations the vector in N Φ(x) Φ(Σ) given byIn the case where m = 3, H(x) is the product of the mean value H = 1/2(κ 1 +κ 2 ) of the principal curvatures κ 1 , κ 2 of the surface at Φ(x) by n, the unit normal vector. Th...

Abstract. Via gauge theory, we give a new proof of partial regularity for harmonic maps in dimensions m ≥ 3 into arbitrary targets. This proof avoids the use of adapted frames and permits to consider targets of "minimal" C 2 regularity. The proof we present moreover extends to a large class of elliptic systems of quadratic growth.

Abstract.We study a two-dimensional model for micromagnetics, which consists in an energy functional over S 2 -valued vector fields. Bounded-energy configurations tend to be planar, except in small regions which can be described as vortices (Bloch lines in physics). As the characteristic "exchangelength" tends to 0, they converge to planar divergence-free unit norm vector fields which jump along line singularities. We derive lower bounds for the energy, which are explicit functions of the jumps of the limit. These lower bounds are proved to be optimal and are achieved by one-dimensional profiles, corresponding to Néel walls, if the jump is small enough (less than π/2 in angle), and by two-dimensional profiles, corresponding to cross-tie walls, if the jump is bigger. Thus, it provides an example of a vector-valued phase-transition type problem with an explicit non-one-dimensional energy-minimizing transition layer. We also establish other lower bounds and compactness properties on different quantities which provide a good notion of convergence and cost of vortices.Mathematics Subject Classification. 35J20, 35J60, 35Q60, 49S05, 49K20.

: We prove a bubble-neck decomposition together with an energy quantization result for sequences of Willmore surfaces into R m with uniformly bounded energy and non-degenerating conformal type. We deduce the strong compactness of Willmore closed surfaces of a given genus modulo the Möbius group action, below some energy threshold. I IntroductionLet Φ be an immersion from a closed abstract two-dimensional manifold Σ into R m≥3 . We denote by g := Φ * g R m the pull back by Φ of the flat canonical metric g R m of R m , also called the first fundamental form of Φ, and we let dvol g be its associated volume form. The Gauss map of the immersion Φ is the map taking values in the Grassmannian of oriented m − 2-planes in R m given bywhere ⋆ is the usual Hodge star operator in the Euclidean metric.Denoting by π n Φ the orthonormal projection of vectors in R m onto the m − 2-plane given by n Φ , the second fundamental form may be expressed asThe mean curvature vector of the immersion at p is H Φ := 1 2 tr g ( I) = 1 2 I(ε 1 , ε 1 ) + I(ε 2 , ε 2 ) , where (ε 1 , ε 2 ) is an orthonormal basis of T p Σ for the metric g Φ .In the present paper, we study the Lagrangian given by the L 2 -norm of the second fundamental form:An elementary computation gives 1The energy E may accordingly be seen as the Dirichlet Energy of the Gauss map n Φ with respect to the induced metric g Φ . The Gauss Bonnet theorem implies thatwhere K Φ is the Gauss curvature of the immersion, and χ(Σ) is the Euler characteristic of the surface Σ. The energyis called Willmore energy.Critical points of the Willmore energy, comprising for example minimal surfaces 2 , are called Willmore surfaces. Although already known in the XIXth century in the context of the elasticity theory of plates, it was first considered in conformal geometry by Blaschke in [Bla3] who sought to merge the theory of minimal surfaces and the conformal invariance property. This Lagrangian has indeed both desired features : its critical points contain minimal surfaces, and it is conformal invariant, owing to the following pointwise identity which holds for an arbitrary immersion Φ of Σ into R m and at every point of Σ :Using again Gauss Bonnet theorem, the latter implies the conformal invariance of W :This conformal invariance implies that the image of a Willmore immersion by a conformal transformation of R m is still a Willmore immersion. Starting for example from a minimal surface, one may then generate many new Willmore surfaces, simply by applying conformal transformations (naturally, these surfaces need no longer be minimal). In his time, Blaschke used the term conformal minimal for the critical points of W , seeking to insist on this idea of merging together the theory of minimal surface with conformal invariance.An important task in the analysis of Willmore surfaces is to understand the closure of the space of Willmore immersions under a certain level of energy. Because of the non-compactness of the conformal group of transformation of R m , one cannot expect that the spa...

: In this work we present new fundamental tools for studying the variations of the Willmore functional of immersed surfaces into R m . This approach gives for instance a new proof of the existence of a Willmore minimizing embedding of an arbitrary closed surface in arbitrary codimension. We explain how the same approach can solve constraint minimization problems for the Willmore functional. We show in particular that, for a given closed surface and a given conformal class for this surface, there is an immersion in R m , away possibly from isolated branched points, which minimizes the Willmore energy among all possible Lipschitz immersions in R m having an L 2 −bounded second fundamental form and realizing this conformal class. This branched immersion is either a smooth Conformal Willmore branched immersion or an isothermic branched immersion. We show that branched points do not exist whenever the minimal energy in the conformal class is less than 8π and that these immersions extend to smooth conformal Willmore embeddings or global isothermic embeddings of the surface in that case. Finally, as a by-product of our analysis, we establish that inside a compact subspace of the moduli space the following holds : weak limit of Palais Smale Willmore sequences are Conformal Willmore, that weak limits of Palais Smale sequences of Conformal Willmore are either Conformal Willmore or Global Isothermic and finally we observe also that weakly converging Palais Smale sequences of Global Isothermic Immersions are Global Isothermic. The analysis developped along the paper -in particular these last results -opens the door to the possibility of constructing new critical saddle points of the Willmore functional without or with constraints using min max methods. I IntroductionThe goal of the present paper is to present a suitable framework to proceed to the calculus of variation of the Willmore functional for immersions.Let Φ be a smooth immersion (rank d Φ is equal to two at every point) from a closed oriented smooth 2-manifold Σ into an euclidian space R m . The first fundamental form g Φ defined by this immersion on Σ is the pull-back by Φ of the metric induced by the restriction of the canonical metric of R m , g R m , to the tangent planes Φ * T Σ of the immersed surface :If there is no ambiguity we simply write g instead of g Φ . We shall denote by n Φ the Gauss map of the immersion Φ which to a point p ∈ Σ assigns the oriented orthonormal (m − 2)−plane to the tangent plane Φ * T p Σ of the immersed surface at Φ(p). n Φ will be seen as a map into the GrassmanianG m−2 (R m ) (or

We consider nonlocal linear Schrödinger-type critical systems of the type(1)where Ω is antisymmetric potential in L 2 (IR, so(m)), v is a IR m valued map and Ω v denotes the matrix multiplication. We show that every solution v ∈ L 2 (IR, IR m ) of (1) is in fact in L p loc (IR, IR m ), for every 2 ≤ p < +∞, in other words, we prove that the system (1) which is a-priori only critical in L 2 happens to have a subcritical behavior for antisymmetric potentials. As an application we obtain the C 0,α loc regularity of weak 1/2-harmonic maps into C 2 compact manifold without boundary.

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