Analysis aspects of Willmore surfaces

Abstract

: 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...

Variational principles for immersed surfaces with L2 -bounded second fundamental form

“…In [Ri2] we explained why the Euler Lagrange written in the form (I.13) seems not compatible from the lagrangian it is coming from in the sense that W ( Φ) only controls the L 2 −norm of the mean curvature whereas in order to give a distributional meaning to the non linearities in the equation like | H| 2 H one needs more information on the regularity of Φ ( H ∈ L 3 for instance for this term). 8 One of the main achievement in [Ri2] was to find a new formulation of the Willmore equation as a conservation law which makes sense for immersions in E Σ .…”

“…8 One of the main achievement in [Ri2] was to find a new formulation of the Willmore equation as a conservation law which makes sense for immersions in E Σ .…”

Local Palais–Smale sequences for the Willmore functional

“…Yet more generally, it is interesting to investigate Palais-Smale sequences. We open our considerations with an "empirical" observation which will hopefully convince the reader that the results derived in [24] offer a suitable framework to acquire information on Palais-Smale sequences of Willmore surfaces. Let us consider a conformal weak Willmore immersion Φ from the flat disk D 2 into R m with square-integrable second fundamental form.…”

“…In codimension 1, there is one equation for the scalar curvature; in higher codimension, the situation becomes significantly more complicated, and one must seek different techniques to approach (1.2). Fortunately, in a conformal parametrization, the second authors showed that it is possible 2 to recast the system (1.2) in an equivalent, but analytically more suitable form [24]. Namely, there holds…”

Erratum to: Sufficient conditions for Willmore immersions in $$\mathbb{R }^3$$ R 3 to be minimal surfaces

“…Having introduced conformal parameters on the disc B R -being equipped with the pull-back metric of X -by means of the general, Riemannian version of Koebe's Mapping Theorem in [1], we may assume that X | B R is already conformally parametrized. Then, Riviere's article [2] shows that X is in fact a real analytic mapping. Moreover, if X is conformally parametrized, then the Willmore equation satisfied by X simplifies to…”