Willmore Type Research Articles

On the conditional existence of foliations by CMC and Willmore type half-spheres

We study half-spheres with small radii [Formula: see text] sitting on the boundary of a smooth bounded domain while meeting it orthogonally. Even though it is known that there exist families of CMC and Willmore type half-spheres near a nondegenerate critical point [Formula: see text] of the domains boundaries mean curvature, it is unknown in both cases whether these provide a foliation of any deleted neighborhood of [Formula: see text]. We prove that this is not guaranteed and establish a criterion in terms of the boundaries geometry that ensures or prevents the respective surfaces from providing such a foliation. This perhaps surprising phenomenon of conditional foliations is absent in the closely related Riemannian setting, where a foliation is guaranteed. We show how this unconditional foliation arises from symmetry considerations and how these fail to apply to the “domain-setting”.

The lifts of surfaces in neutral 4-manifolds into the 2-Grassmann bundles

A twistor lift of a space-like or time-like surface in a neutral hyperKähler 4-manifold with zero mean curvature vector is given by a (para)holomorphic function, which yields (para)holomorphicity of the Gauss maps of space-like or time-like surfaces in E24 with zero mean curvature vector. For a space-like or time-like surface in an oriented neutral 4-manifold with zero mean curvature vector such that both twistor lifts belong to the kernel of the curvature tensor, its (para)complex quartic differential is holomorphic. If both twistor lifts of a time-like surface with zero mean curvature vector have light-like or zero covariant derivatives, then either the shape operator with respect to a light-like normal vector field vanishes or all the shape operators of the surface are light-like or zero. Examples with the former (resp. latter) property are given by the conformal Gauss maps of time-like surfaces of Willmore type with zero paraholomorphic quartic differential (resp. time-like surfaces in 4-dimensional neutral space forms based on the Gauss-Codazzi-Ricci equations).

Refined position estimates for surfaces of Willmore type in Riemannian manifolds

Refined position estimates for surfaces of Willmore type in Riemannian manifolds

Local foliation of manifolds by surfaces of Willmore type

We show the existence of a local foliation of a three dimensional Riemannian manifold by critical points of the Willmore functional subject to a small area constraint around non-degenerate critical points of the scalar curvature. This adapts a method developed by Rugang Ye to construct foliations by surfaces of constant mean curvature.

Long time existence of solutions to an elastic flow of networks

The L2-gradient flow of the elastic energy of networks leads to a Willmore type evolution law with non-trivial nonlinear boundary conditions. We show local in time existence and uniqueness for this elastic flow of networks in a Sobolev space setting under natural boundary conditions. In addition, we show a regularisation property and geometric existence and uniqueness. The main result is a long time existence result using energy methods.

Surfaces with zero mean curvature vector in neutral 4-manifolds

Space-like surfaces and time-like surfaces with zero mean curvature vector in oriented neutral 4-manifolds are isotropic and compatible with the orientations of the spaces if and only if their lifts to the space-like and the time-like twistor spaces respectively are horizontal. In neutral Kähler surfaces and paraKähler surfaces, complex curves and paracomplex curves respectively are such surfaces and characterized by one additional condition. In neutral 4-dimensional space forms, the holomorphic quartic differentials defined on such surfaces vanish. There exist time-like surfaces with zero mean curvature vector and zero holomorphic quartic differential which are not compatible with the orientations of the spaces and the conformal Gauss maps of time-like surfaces of Willmore type and their analogues give such surfaces.

The Area Preserving Willmore Flow and Local Maximizers of the Hawking Mass in Asymptotically Schwarzschild Manifolds

We study the area preserving Willmore flow in an asymptotic region of an asymptotically flat manifold which is C^3-close to Schwarzschild. It was shown by Lamm, Metzger and Schulze that such a region is foliated by spheres of Willmore type, see (Lamm et al. in Math Ann 350(1):1–78, 2011). In this paper, we prove that the leaves of this foliation are stable under small area preserving W^{2,2}-perturbations with respect to the area preserving Willmore flow. This implies, in particular, that the leaves are strict local area preserving maximizers of the Hawking mass with respect to the W^{2,2}-topology.

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.

Elastic energy of a convex body

In this paper a Blaschke‐Santaló diagram involving the area, the perimeter and the elastic energy of planar convex bodies is considered. More precisely we give a description of set urn:x-wiley:0025584X:media:mana201400256:mana201400256-math-0001where A is the area, P is the perimeter and E is the elastic energy, that is a Willmore type energy in the plane. In order to do this, we investigate the following shape optimization problem: urn:x-wiley:0025584X:media:mana201400256:mana201400256-math-0002where is the class of convex bodies with fixed perimeter and is a parameter. Existence, regularity and geometric properties of solutions to this minimum problem are shown.

Variational problem of an abstract Willmore type functional of sub-manifold in unit sphere

For an n -dimensional sub-manifold in unit sphere, we introduce an abstract Willmore type called W ( n,F ) -Willmore functional, which generalizes the well-known classic Willmore functional. Its critical point is called the W ( n,F ) -Willmore sub-manifold, for which the variational equation and the Simons' type integral inequalities are obtained. Moreover, for some particular functions F , we construct a few examples of W ( n,F ) - Willmore sub-manifold. What's more important, a gap phenomenon characterization is given by use of Simons' type integral formula.

Bubble tree of branched conformal immersions and applications to the Willmore functional

We develop a bubble tree construction and prove compactness results for $W^{2,2}$ branched conformal immersions of closed Riemann surfaces, with varying conformal structures whose limit may degenerate, in ${\Bbb R}^n$ with uniformly bounded areas and Willmore energies. The compactness property is applied to construct Willmore type surfaces in compact Riemannian manifolds. This includes (a) existence of a Willmore $2$-sphere in ${\Bbb S}^n$ with at least 2 nonremovable singular points (b) existence of minimizers of the Willmore functional with prescribed area in a compact manifold $N$ provided (i) the area is small when genus is $0$ and (ii) the area is close to that of the area minimizing surface of Schoen-Yau and Sacks-Uhlenbeck in the homotopy class of an incompressible map from a surface of positive genus to $N$ and $\pi_2(N)$ is trivial (c) existence of smooth minimizers of the Willmore functional if a Douglas type condition is satisfied.

Elastic energy of a convex body

In this paper a Blaschke-Santalo diagram involving the area, the perimeter and the elastic energy of planar convex bodies is considered. More precisely we give a description of set $$\mathcal{E}:=\left\{(x,y)\in \R^2, x=\frac{4\pi A(\Omega)}{P(\Omega)^2},y=\frac{E(\Omega)P(\Omega)}{2\pi^2},\,\Omega\mbox{convex} \right\},$$ where $A$ is the area, $P$ is the perimeter and $E$ is the elastic energy, that is a Willmore type energy in the plane. In order to do this, we investigate the following shape optimization problem: $$\min_{\Omega\in\mathcal{C}}\{E(\Omega)+\mu A(\Omega)\},$$ where $\mathcal{C}$ is the class of convex bodies with fixed perimeter and $\mu\ge 0$ is a parameter. Existence, regularity and geometric properties of solutions to this minimum problem are shown.

Gap phenomenon of an abstract Willmore type functional of hypersurface in unit sphere.

For an n-dimensional hypersurface in unit sphere, we introduce an abstract Willmore type called W (n,F)-Willmore functional, which generalizes the well-known classic Willmore functional. Its critical point is called the W (n,F)-Willmore hypersurface, for which the variational equation and Simons' type integral equalities are obtained. Moreover, we construct a few examples of W (n,F)-Willmore hypersurface and give a gap phenomenon characterization by use of our integral formula.

Minimizers of the Willmore functional with a small area constraint

We show the existence of a smooth spherical surface minimizing the Willmore functional subject to an area constraint in a compact Riemannian three-manifold, provided the area is small enough. Moreover, we partially classify complete surfaces of Willmore type with positive mean curvature in Riemannian three-manifolds.

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.

Small Surfaces of Willmore Type in Riemannian Manifolds

In this paper we investigate the properties of small surfaces of Willmore type in Riemannian manifolds. By small surfaces we mean topological spheres contained in a geodesic ball of small enough radius. In particular, we show that if there exist such surfaces with positive mean curvature in the geodesic ball Br(p) for arbitrarily small radius r around a point p in the Riemannian manifold, then the scalar curvature must have a critical point at p.

Inequalities of Willmore type for submanifolds

Inequalities of Willmore type for submanifolds