Energy quantization for Willmore surfaces and applications

Abstract

: 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...

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