Analysis of constrained Willmore surfaces

This paper investigates the regularity of constrained Willmore immersions into ?^m≥3 locally around both “regular” points and around branch points, where the immersive nature of the map degenerates. We develop local asymptotic expansions for the immersion and its first and second derivatives, given in terms of residues computed as circulation integrals. We deduce explicit “point removability” conditions ensuring that the immersion is smooth. Our results apply in particular to Willmore immersions and to parallel mean curvature immersions in any codimension.     

Sequences of Willmore surfaces

In this paper we develop the theory of Willmore sequences for Willmore surfaces in the 4-sphere. We show that under appropriate conditions this sequence has to terminate. In this case the Willmore surface either is the Twistor projection of a holomorphic curve into or the inversion of a minimal surface with planar ends in . These results give a unified explanation of previous work on the characterization of Willmore spheres and Willmore tori with non-trivial normal bundles by various authors. K. Leschke thanks the Department of Mathematics and Statistics at the University of Massachusetts, Amherst, and the Center for Geometry, Analysis, Numerics and Graphics for their support and hospitality.     

Willmore surfaces in the four-sphere

Willmore immersions of an orientable surface X in the n-dimensionalsphere appear as the extremal points of a conformally invariant variational problem in the space of all immersions f: X S ⁿ.In this paper we will study Willmore immersions of the differentiable two-sphere in S ⁴, using the method of moving frames and Cartan's conformal structures.The work on this paper was partially supported by a Fellowship of the Consiglio Nazionale delle Ricerche.     

The Willmore functional of surfaces

The Whitney immersion theorem asserts that every smooth n-dimensional manifold can be immersed in R2n-1, in particular, smooth surfaces can be immersed in R3, and for embeddings the ambient dimension needs to go up by 1. Immersions or embeddings carry both intrinsic and extrinsic information of manifolds, and the latter determines how the manifold fits into the Euclidean space. The key geometric quantity to capture the extrinsic geometry is the second fundamental form.     

Equivariant constrained Willmore tori in the 3-sphere

In this paper we study equivariant constrained Willmore tori in the 3-sphere. These tori admit a 1-parameter group of Möbius symmetries and are critical points of the Willmore energy under conformal variations. We show that the spectral curve associated to an equivariant torus is given by a double covering of \(\mathbb {C}\) and classify equivariant constrained Willmore tori by the genus \(g\) of their spectral curve. In this case the spectral genus satisfies \(g \le 3\) .     

On Willmore Legendrian surfaces in $$\mathbb {S}^5$$ and the contact stationary Legendrian Willmore surfaces

In this paper we study Willmore Legendrian surfaces (that is Legendrian surfaces which are critical points of the Willmore functional). We use an equality proved in Luo (arXiv:1211.4227v6) to get a relation between Willmore Legendrian surfaces and contact stationary Legendrian surfaces in \(\mathbb {S}^5\), and then we use this relation to prove a classification result for Willmore Legendrian spheres in \(\mathbb {S}^5\). We also get an integral inequality for Willmore Legendrian surfaces and in particular we prove that if the square length of the second fundamental form of a Willmore Legendrian surface in \(\mathbb {S}^5\) belongs to [0, 2], then it must be 0 and L is totally geodesic or 2 and L is a flat minimal Legendrian tori, which generalizes the result of Yamaguchi et al. (Proc Am Math Soc 54:276–280, 1976). We also study variation of the Willmore functional among Legendrian surfaces in 5-dimensional Sasakian manifolds. Let \(\Sigma \) be a closed surface and \((M,\alpha ,g_\alpha ,J)\) a 5-dimensional Sasakian manifold with a contact form \(\alpha \), an associated metric \(g_\alpha \) and an almost complex structure J. Assume that \(f:\Sigma \mapsto M\) is a Legendrian immersion. Then f is called a contact stationary Legendrian Willmore surface (in short, a csL Willmore surface) if it is a critical point of the Willmore functional under contact deformations. To investigate the existence of csL Willmore surfaces we introduce a higher order flow which preserves the Legendre condition and decreases the Willmore energy. As a first step we prove that this flow is well posed if \((M,\alpha ,g_\alpha ,J)\) is a Sasakian Einstein manifold, in particular \(\mathbb {S}^5\).     

Analysis aspects of Willmore surfaces

A new formulation for the Euler–Lagrange equation of the Willmore functional for immersed surfaces in ℝ ^m is given as a nonlinear elliptic equation in divergence form, with non-linearities comprising only Jacobians. Letting be the mean curvature vector of the surface, our new formulation reads , where is a well-defined locally invertible self-adjoint elliptic operator. Several consequences are studied. In particular, the long standing open problem asking for a meaning to the Willmore Euler–Lagrange equation for immersions having only L ²-bounded second fundamental form is now solved. The regularity of weak Willmore immersions with L ²-bounded second fundamental form is also established. Its proof relies on the discovery of conservation laws which are preserved under weak convergence. A weak compactness result for Willmore surfaces with energy less than 8π (the Li–Yau condition ensuring the surface is embedded) is proved, via a point removability result established for Wilmore surfaces in ℝ ^m , thereby extending to arbitrary codimension the main result in [KS3]. Finally, from this point-removability result, the strong compactness of Willmore tori below the energy level 8π is proved both in dimension 3 (this had already been settled in [KS3]) and in dimension 4.     

Envelopes and osculates of Willmore surfaces

We view conformal surfaces in the 4-sphere as quaternionic holomorphiccurves in quaternionic projective space. By constructing envelopingand osculating curves, we obtain new holomorphic curves in quaternionicprojective space and thus new conformal surfaces. Applying theseconstructions to Willmore surfaces, we show that the osculatingand enveloping curves of Willmore spheres remain Willmore.     

Adjoint transform of Willmore surfaces in

Using moving frame method, we study the Möbius geometry of a pair of conformally immersed surfaces in . Two new invariants θ and ρ associated with them arise naturally as well as the notion of touch and co-touch. As an application, adjoint transform is defined for any Willmore surface in . It always exists locally, hence generalizes known duality theorems of Willmore surfaces. Finally we characterize a pair of adjoint Willmore surfaces in terms of harmonic map.     

Symmetric Willmore surfaces of revolution satisfying natural boundary conditions

We consider the Willmore-type functional $$\mathcal{W}_{\gamma}(\Gamma):= \int\limits_{\Gamma} H^2 \; dA -\gamma \int\limits_{\Gamma} K \; dA,$$ where H and K denote mean and Gaussian curvature of a surface Γ, and ${\gamma \in [0,1]}$ is a real parameter. Using direct methods of the calculus of variations, we prove existence of surfaces of revolution generated by symmetric graphs which are solutions of the Euler-Lagrange equation corresponding to ${\mathcal{W}_{\gamma}}$ and which satisfy the following boundary conditions: the height at the boundary is prescribed, and the second boundary condition is the natural one when considering critical points where only the position at the boundary is fixed. In the particular case γ = 0 these boundary conditions are arbitrary positive height α and zero mean curvature.     

Foliations of asymptotically flat manifolds by surfaces of Willmore type

The goal of this paper is to establish the existence of a foliation of the asymptotic region of an asymptotically flat manifold with positive mass by surfaces which are critical points of the Willmore functional subject to an area constraint. Equivalently these surfaces are critical points of the Geroch–Hawking mass. Thus our result has applications in the theory of general relativity.     

Unstable Willmore surfaces of revolution subject to natural boundary conditions

In the class of surfaces with fixed boundary, critical points of the Willmore functional are naturally found to be those solutions of the Euler-Lagrange equation where the mean curvature on the boundary vanishes. We consider the case of symmetric surfaces of revolution in the setting where there are two families of stable solutions given by the catenoids. In this paper we demonstrate the existence of a third family of solutions which are unstable critical points of the Willmore functional, and which spatially lie between the upper and lower families of catenoids. Our method does not require any kind of smallness assumption, and allows us to derive some additional interesting qualitative properties of the solutions.     

On Coverings of Tori with Cubes

Doklady Mathematics - We obtain new bounds and exact values of the minimum number of cubes with side length $$\varepsilon \in (0,\;1)$$ covering the torus...