Willmore Surfaces in S n

A surface x> : M S ⁿ is called a Willmore surface if it is a critical surface of the Willmore functional. It is well known that any minimal surface is a Willmore surface and that many nonminimal Willmore surfaces exists. In this paper, we establish an integral inequality for compact Willmore surfaces in S ⁿ and obtain a new characterization of the Veronese surface in S ⁴ as a Willmore surface. Our result reduces to a well-known result in the case of minimal surfaces.     

On a Generalization of Willmore Surfaces for Hypersurfaces

This paper gives a generalization of Willmore surfaces for hypersurfaces in higher dimensional space, based on a property of central hyperspheres which was already discovered by Blaschke and Thomsen in the case of surfaces.     

New Examples of Willmore Surfaces in S n

A surface x: M S ⁿ is called a Willmore surface if it is a criticalsurface of the Willmore functional _M (S – 2H ²)dv, where H isthe mean curvature and S is the square of the length of the secondfundamental form. It is well known that any minimal surface is aWillmore surface. The first nonminimal example of a flat Willmoresurface in higher codimension was obtained by Ejiri. This example whichcan be viewed as a tensor product immersion of S ¹(1) and a particularsmall circle in S ²(1), and therefore is contained in S ⁵(1) gives anegative answer to a question by Weiner. In this paper we generalize theabove mentioned example by investigating Willmore surfaces in S ⁿ (1)which can be obtained as a tensor product immersion of two curves. We inparticular show that in this case too, one of the curves has to beS ¹(1), whereas the other one is contained either in S ²(1) or in S ³(1). In the first case, we explicitly determine the immersion interms of elliptic functions, thus constructing infinetely many newnonminimal flat Willmore surfaces in S ⁵. Also in the latter casewe explicitly include examples.     

Willmore Surfaces of Revolution with Two Prescribed Boundary Circles

We consider the family of smooth embedded surfaces of revolution in ?³ having two concentric circles contained in two parallel planes of ?³ as boundary. Minimizing the Willmore functional within this class of surfaces we prove the existence of smooth axi-symmetric Willmore surfaces having these circles as boundary. When the radii of the circles tend to zero we prove convergence of these solutions to the round sphere.     

Nonuniqueness for Willmore Surfaces of Revolution Satisfying Dirichlet Boundary Data

In this note Willmore surfaces of revolution with Dirichlet boundary conditions are considered. We show two nonuniqueness results by reformulating the problem in the hyperbolic half plane and solving a suitable initial value problem for the corresponding elastic curves. The behavior of such elastic curves is examined by a method provided by Langer and Singer to reduce the order of the underlying ordinary differential equation. This ensures that these solutions differ from solutions already obtained by Dall’Acqua, Deckelnick and Grunau. We will additionally provide a Bernstein-type result concerning the profile curve of a Willmore surface of revolution. If this curve is a graph on the whole real numbers it has to be a Möbius transformed catenary. We show this by a corollary of the above-mentioned method by Langer and Singer.     

Conformal Gauss Map Geometry and Application to Willmore Surfaces in Model Spaces

Potential Analysis - In this paper we make a detailed and self-contained study of the conformal Gauss map. Then, starting from the seminal work of Bryant (J. Differential Geom. 20(1), 23–53...     

Surfaces of revolution in the Heisenberg group and the spectral generalization of the Willmore functional

We study the generalization of the Willmore functional for surfaces in the three-dimensional Heisenberg group. Its construction is based on the spectral theory of the Dirac operator entering into theWeierstrass representation of surfaces in this group. Using the surfaces of revolution we demonstrate that the generalization resembles the Willmore functional for the surfaces in the Euclidean space in many geometrical aspects. We also observe the relation of these functionals to the isoperimetric problem.     

Constrained Willmore surfaces

Constrained Willmore surfaces are conformal immersions of Riemann surfaces that are critical points of the Willmore energy under compactly supported infinitesimal conformal variations. Examples include all constant mean curvature surfaces in space forms. In this paper we investigate more generally the critical points of arbitrary geometric functionals on the space of immersions under the constraint that the admissible variations infinitesimally preserve the conformal structure. Besides constrained Willmore surfaces we discuss in some detail examples of constrained minimal and volume critical surfaces, the critical points of the area and enclosed volume functional under the conformal constraint. C. Bohle, G. P. Peters and U. Pinkall are partially supported by DFG SPP 1154.     

Computational parametric Willmore flow

We propose a new algorithm for the computation of Willmore flow. This is the L ²-gradient flow for the Willmore functional, which is the classical bending energy of a surface. Willmore flow is described by a highly nonlinear system of PDEs of fourth order for the parametrization of the surface. The spatially discrete numerical scheme is stable and consistent. The discretization relies on an adequate calculation of the first variation of the Willmore functional together with a derivation of the second variation of the area functional which is well adapted to discretization techniques with finite elements. The algorithm uses finite elements on surfaces. We give numerical examples and tests for piecewise linear finite elements. A convergence proof for the full algorithm remains an open question.     

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.     

The Willmore boundary problem

We consider the Euler–Lagrange equation of the Willmore functional coupled with Dirichlet and Neumann boundary conditions on a given curve. We prove existence of a branched solution, and for Willmore energy < 8π, we prove the existence of a smooth, embedded solution.     

Towards the Willmore conjecture

We develop a variety of approaches, mainly using integral geometry, to proving that the integral of the square of the mean curvature of a torus immersed in must always take a value no less than . Our partial results, phrased mainly within the -formulation of the problem, are typically strongest when the Gauss curvature can be controlled in terms of extrinsic curvatures or when the torus enjoys further properties related to its distribution within the ambient space (see Sect. 3). Corollaries include a recent result of Ros [20] confirming the Willmore conjecture for surfaces invariant under the antipodal map, and a strengthening of the expected results for flat tori. The value arises in this work in a number of different ways – as the volume (or renormalised volume) of or , and in terms of the length of shortest nontrivial loops in subgroups of SO(4). Received April 26, 1999 / Accepted January 14, 2000 / Published online June 28, 2000     

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.     

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.     

Willmore two-spheres in the four-sphere

Genus zero Willmore surfaces immersed in the three-sphere correspond via the stereographic projection to minimal surfaces in Euclidean three-space with finite total curvature and embedded planar ends. The critical values of the Willmore functional are , where , with . When the ambient space is the four-sphere , the regular homotopy class of immersions of the two-sphere is determined by the self-intersection number ; here we shall prove that the possible critical values are , where . Moreover, if , the corresponding immersion, or its antipodal, is obtained, via the twistor Penrose fibration , from a rational curve in and, if , via stereographic projection, from a minimal surface in with finite total curvature and embedded planar ends. An immersion lies in both families when the rational curve is contained in some or (equivalently) when the minimal surface of is complex with respect to a suitable complex structure of .

     

Willmore flow of planar networks

Geometric gradient flows for elastic energies of Willmore type play an important role in mathematics and in many applications. The evolution of elastic curves has been studied in detail both for closed as well as for open curves. Although elastic flows for networks also have many interesting features, they have not been studied so far from the point of view of mathematical analysis. So far it was not even clear what are appropriate boundary conditions at junctions. In this paper we give a well-posedness result for Willmore flow of networks in different geometric settings and hence lay a foundation for further mathematical analysis. A main point in the proof is to check whether different proposed boundary conditions lead to a well posed problem. In this context one has to check the Lopatinskii–Shapiro condition in order to apply the Solonnikov theory for linear parabolic systems in Hölder spaces which is needed in a fixed point argument. We also show that the solution we get is unique in a purely geometric sense.     

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.     

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.     

Cahn-Hilliard inpainting and the Willmore functional

The Cahn-Hilliard equation has its origin in material sciences and serves as a model for phase separation and phase coarsening in binary alloys. A new approach in the class of fourth order inpainting algorithms is inpainting of binary images using the Cahn-Hilliard equation. Like solutions of the Cahn-Hilliard equation converging to two main values during the phase separation process, the grayvalues inside the missing part of the image are oriented towards the binary states black and white. We present stability/instability results for solutions of the Cahn-Hilliard equation and their connection to the Willmore functional. In particular we will consider the Willmore functional as a quantity to find the optimal scale of the inpainting result. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)