: 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...

I OverviewThe Willmore energy of an immersed closed surface Φ : Σ → R m≥3 is given bywhere H denotes the weak mean curvature vector, and dvol g is the area form of the metric g induced on Φ(Σ) by the canonical Euclidean metric on R m . Critical points of the Lagrangian W for perturbations of the form Φ+t ξ, where ξ is an arbitrary compactly supported smooth map on Σ into R m , are known as Willmore surfaces. Not only is the Willmore functional invariant under reparametrization, but more importantly, it is invariant under the group of Möbius transformations of R m ∪{∞}. This remarkable property prompts the use of the Willmore energy in various fields of science. A survey of the Willmore functional, of its properties, and of the relevant literature is available in [Ri3]. Owing to the Gauss-Bonnet theorem, we note that the Willmore energy (I.1) may be equivalently expressed aswhere I is the second fundamental form, and χ(Σ) is the Euler characteristic of Σ, which is a topological invariant for a closed surface. From the variational point of view, Willmore surfaces are thus critical points of the energyIt then appears natural to restrict our attention on immersions whose second fundamental forms are locally square-integrable.We assume that the point-singularity lies at the origin, and we localize the problem by considering a map Φ : D 2 → R m≥3 , which is an immersion of D 2 \ {0}, and satisfyingBy a procedure detailed in [KS2], it is possible to construct a parametrization ζ of the unit-disk such that Φ•ζ is conformal. To do so, one first extends Φ to all of C\{0} while keeping a bounded image and the second fundamental form squareintegrable. One then shifts so as to have Φ(0) = 0, and inverts about the origin so as to obtain a complete immersion with square-integrable second fundamental form. Calling upon a result of Huber [Hu] (see also [MS] and [To]), one deduces that the image of the immersion is conformally equivalent to C. Inverting yet once more about the origin finally gives the desired conformal immersion 1 , which we shall abusively continue to denote Φ. It has the aforementioned properties (i)-(iii), and moreover,1 which degenerates at the origin in a particular way, see (I.9).Away from the origin, we define the Gauss map n viawhere (x 1 , x 2 ) are standard Cartesian coordinates on the unit-disk D 2 , and ⋆ is the Euclidean Hodge-star operator. The immersion Φ is conformal, i.e.where λ is the conformal parameter. An elementary computation shows thatHence, by hypothesis, we see that n ∈ W 1,2 (D 2 \ {0}). In dimension two, the 2-capacity of isolated points is null, so we actually have Φ ∈ W 1,2 (D 2 ) and n ∈ W 1,2 (D 2 ) (note however that Φ remains a non-degenerate immersion only away from the singularity). Rescaling if necessary, we shall henceforth always assume thatwhere the adjustable parameter ε 0 is chosen to fit our various needs (in particular, we will need it to be "small enough" in Proposition A.1).For the sake of the following paragraph, we consider a conformal immersion Φ : D 2 → R m , whi...

Noether's theorem and the invariances of the Willmore functional are used to derive conservation laws that are satis ed by the critical points of the Willmore energy subject to generic constraints. We recover in particular previous results independently obtained by R. Capovilla and J. Guven, and by T. Rivière. Several examples are considered in detail.

Abstract. We derive the Euler-Lagrange equations for minimizers of causal variational principles in the non-compact setting with constraints, possibly prescribing symmetries. Considering first variations, we show that the minimizing measure is supported on the intersection of a hyperplane with a level set of a function which is homogeneous of degree two. Moreover, we perform second variations to obtain that the compact operator representing the quadratic part of the action is positive semi-definite. The key ingredient for the proof is a subtle adaptation of the Lagrange multiplier method to variational principles on convex sets.

Using the reformulation in divergence form of the Euler-Lagrange equation for the Willmore functional as it was developed in the second author's paper [24], we study the limit of a local PalaisSmale sequence of weak Willmore immersions with locally squareintegrable second fundamental form. We show that the limit immersion is smooth and that it satisfies the conformal Willmore equation: it is a critical point of the Willmore functional restricted to infinitesimal conformal variations.