Willmore Surfaces Research Articles

Large area-constrained Willmore surfaces in asymptotically Schwarzschild $3$-manifolds

Large area-constrained Willmore surfaces in asymptotically Schwarzschild $3$-manifolds

Polynomial conserved quantities for constrained Willmore surfaces

Polynomial conserved quantities for constrained Willmore surfaces

Willmore flow of complete surfaces

We consider the Willmore flow equation for complete, properly immersed surfaces in Rn. Given bounded geometry on the initial surface, we extend the result in [5] and prove short time existence and uniqueness of the Willmore flow. We also show that a complete Willmore surface with low Willmore energy must be a plane, and that a Willmore flow with low initial energy and Euclidean volume growth must converge smoothly to a plane.

On the Index of Willmore spheres

We consider unbranched Willmore surfaces in Euclidean space that arise as inverted complete minimal surfaces with embedded planar ends. Several statements are proven about upper and lower bounds on the Morse Index—the number of linearly independent variational directions that locally decrease the Willmore energy. We in particular compute the Index of a Willmore sphere in the three‑space. This Index is $m-d$, where $m$ is the number of ends of the corresponding complete minimal surface and $d$ is the dimension of the span of the normals at the $m$-fold point. The dimension $d$ is either two or three. For $m=4$, we prove that $d=3$. In general, we show that there is a strong connection of the Morse Index to the number of logarithmically growing Jacobi fields on the corresponding minimal surface.

Classification of Willmore Surfaces with Vanishing Gaussian Curvature

Classification of Willmore Surfaces with Vanishing Gaussian Curvature

Analysis of the inhomogeneous Willmore equation

We study a class of fourth-order geometric problems modeling Willmore surfaces, conformally constrained Willmore surfaces, isoperimetrically constrained Willmore surfaces, and bi-harmonic surfaces in the sense of Chen, among others. We prove several local energy estimates and derive a global gap lemma.

Energy Estimates for the Tracefree Curvature of Willmore Surfaces and Applications

We prove an $$\varepsilon $$ -regularity result for the tracefree curvature of a Willmore surface with bounded second fundamental form. For such a surface, we obtain a pointwise control of the tracefree second fundamental form from a small control of its $$L^2$$ -norm. Several applications are investigated. Notably, we derive a gap statement for surfaces of the aforementioned type. We further apply our results to deduce regularity results for conformal minimal spacelike immersions into the de Sitter space $${\mathbb {S}}^{4,1}$$ .

Local foliations by critical surfaces of the Hawking energy and small sphere limit

Local foliations of area constrained Willmore surfaces on a 3-dimensional Riemannian manifold were constructed by Lamm et al (2020 Ann. Inst. Fourier 70 1639–62) and Ikoma et al (2020 Int. Math. Res. Not. 70 6538–68), the leaves of these foliations are in particular critical surfaces of the Hawking energy in case they are contained in a totally geodesic spacelike hypersurface. We generalize these foliations to the general case of a non-totally geodesic spacelike hypersurface, constructing an unique local foliation of area constrained critical surfaces of the Hawking energy. A discrepancy when evaluating the so called small sphere limit of the Hawking energy was found by Friedrich (2020 arXiv:1909.02388v2 [math.DG]), he studied concentrations of area constrained critical surfaces of the Hawking energy and obtained a result that apparently differs from the well established small sphere limit of the Hawking energy of Horowitz and Schmidt (1982 Proc. R. Soc. A 381 215–24), this small sphere limit in principle must be satisfied by any quasi local energy. We independently confirm the discrepancy and explain the reasons for it to happen. We also prove that these surfaces are suitable to evaluate the Hawking energy in the sense of Lamm et al (2011 Math. Ann. 350 1–78), and we find an indication that these surfaces may induce an excess in the energy measured.

Pointwise Expansion of Degenerating Immersions of Finite Total Curvature

Generalising classical result of Müller and Šverák (J. Differ. Geom. 42(2), 229-258, 1995), we obtain a pointwise estimate of the conformal factor of sequences of conformal immersions from the unit disk of the complex plane of uniformly bounded total curvature and converging strongly outside of a concentration point towards a branched immersions for which the quantization of energy holds. We show that the multiplicity associated to the conformal parameter becomes eventually constant to an integer equal to the order of the branch point of the limiting branched immersion. Furthermore, we deduce a C^0 convergence of the normal unit in the neck regions. Finally, we show that these improved energy quantizations hold for Willmore surfaces of uniformly bounded energy and precompact conformal class, and for Willmore spheres arising as solutions of min-max problems in the viscosity method.

Quartic differentials and harmonic maps in conformal surface geometry

We consider codimension 2 sphere congruences in pseudo-conformal geometry that are harmonic with respect to the conformal structure of an orthogonal surface. We characterise the orthogonal surfaces of such congruences as either $S$-Willmore surfaces, quasi-umbilical surfaces, constant mean curvature surfaces in 3-dimensional space forms or surfaces of constant lightlike mean curvature in 3-dimensional lightcones. We then investigate Bryant's quartic differential in this context and show that generically this is divergence free if and only if the surface under consideration is either superconformal or orthogonal to a harmonic congruence of codimension 2 spheres. We may then apply the previous result to characterise surfaces with such a property.

Weyl's law for the eigenvalues of the Neumann–Poincaré operators in three dimensions: Willmore energy and surface geometry

We deduce eigenvalue asymptotics of the Neumann–Poincaré operators in three dimensions. The region Ω is C2,α (α>0) bounded in R3 and the Neumann–Poincaré operator K∂Ω:L2(∂Ω)→L2(∂Ω) is defined byK∂Ω[ψ](x):=14π∫∂Ω〈y−x,n(y)〉|x−y|3ψ(y)dSy where dSy is the surface element, n(y) is the outer normal vector on ∂Ω, and the integral is understood in the principal value sense. Then the ordered eigenvalues λj(K∂Ω) of the Neumann–Poincaré operator satisfy|λj(K∂Ω)|∼{3W(∂Ω)−2πχ(∂Ω)128π}1/2j−1/2asj→∞. Here W(∂Ω) and χ(∂Ω) denote the Willmore energy and the Euler characteristic of the boundary surface ∂Ω. This formula is the so-called Weyl's law for eigenvalue problems of Neumann–Poincaré operators.

Willmore surfaces and Hopf tori in homogeneous 3-manifolds

Some classification results for closed surfaces in Berger spheres are presented. On the one hand, a Willmore functional for isometrically immersed surfaces into an homogeneous space {mathbb {E}}^{3}(kappa ,tau ) with isometry group of dimension 4 is defined and its first variational formula is computed. Then, we characterize Clifford and Hopf tori as the only Willmore surfaces satisfying a sharp Simons-type integral inequality. On the other hand, we also obtain some integral inequalities for closed surfaces with constant extrinsic curvature in {mathbb {E}}^3(kappa ,tau ), becoming equalities if and only if the surface is a Hopf torus in a Berger sphere.

The Willmore Center of Mass of Initial Data Sets

We refine the Lyapunov–Schmidt analysis from our recent paper (Eichmair and Koerber in Large area-constrained Willmore surfaces in asymptotically Schwarzschild 3-manifolds. arXiv preprint arXiv:2101.12665, 2021) to study the geometric center of mass of the asymptotic foliation by area-constrained Willmore surfaces of initial data for the Einstein field equations. If the scalar curvature of the initial data vanishes at infinity, we show that this geometric center of mass agrees with the Hamiltonian center of mass. By contrast, we show that the positioning of large area-constrained Willmore surfaces is sensitive to the distribution of the energy density. In particular, the geometric center of mass may differ from the Hamiltonian center of mass if the scalar curvature does not satisfy additional asymptotic symmetry assumptions.

The Classification of Branched Willmore Spheres in the 3-Sphere and the 4-Sphere

We extend the classification of Robert Bryant of Willmore spheres in $S^3$ to variational branched Willmore spheres $S^3$ and show that they are inverse stereographic projections of complete minimal surfaces with finite total curvature in $\mathbb{R}^3$ and vanishing flux. We also obtain a classification of variational branched Willmore spheres in $S^4$, generalising a theorem of Sebastian Montiel. As a result of our asymptotic analysis at branch points, we obtain an improved $C^{1,1}$ regularity of the unit normal of variational branched Willmore surfaces in arbitrary codimension. We also prove that the width of Willmore sphere min-max procedures in dimension $3$ and $4$, such as the sphere eversion, is an integer multiple of $4\pi$.

Stability properties of 2-lobed Delaunay tori in the 3-sphere

We show that the 2-lobed Delaunay tori are stable as constrained Willmore surfaces in the 3-sphere. [Display omitted] The images are made by Nick Schmitt.

Constrained willmore surfaces

Constrained willmore surfaces

Constrained willmore surfaces

Smooth curves and surfaces can be characterized as minimizers of squared curvature bending energies subject to constraints. In the univariate case with an isometry (length) constraint this leads to classic non-linear splines. For surfaces, isometry is too rigid a constraint and instead one asks for minimizers of the Willmore (squared mean curvature) energy subject to a conformality constraint. We present an efficient algorithm for (conformally) constrained Willmore surfaces using triangle meshes of arbitrary topology with or without boundary. Our conformal class constraint is based on the discrete notion of conformal equivalence of triangle meshes. The resulting non-linear constrained optimization problem can be solved efficiently using the competitive gradient descent method together with appropriate Sobolev metrics. The surfaces can be represented either through point positions or differential coordinates. The latter enable the realization of abstract metric surfaces without an initial immersion. A versatile toolkit for extrinsic conformal geometry processing, suitable for the construction and manipulation of smooth surfaces, results through the inclusion of additional point, area, and volume constraints.

Willmore Surfaces in Spheres via Loop Groups IV: On Totally Isotropic Willmore Two-Spheres in S6

In this paper the author derives a geometric characterization of totally isotropic Willmore two-spheres in S6, which also yields to a description of such surfaces in terms of the loop group language. Moreover, applying the loop group method, he also obtains an algorithm to construct totally isotropic Willmore two-spheres in S6. This allows him to derive new examples of geometric interests. He first obtains a new, totally isotropic Willmore two-sphere which is not S-Willmore (i.e., has no dual surface) in S6. This gives a negative answer to an open problem of Ejiri in 1988. In this way he also derives many new totally isotropic, branched Willmore two-spheres which are not S-Willmore in S6.

Willmore minmax surfaces and the cost of the sphere eversion

We develop a general minmax procedure in Euclidian spaces for constructing Willmore surfaces of non-zero indices. We apply this procedure to the Willmore minmax sphere eversion in the 3-dimensional Euclidian space. We compute the cost of sphere eversion in terms of Willmore energies of the Willmore spheres in \mathbb R^3 .

Classification of homogeneous Willmore surfaces in $S^n$

In this note we consider homogeneous Willmore surfaces in $S^{n+2}$. The main result is that a homogeneous Willmore two-sphere is conformally equivalent to a homogeneous minimal two-sphere in $S^{n+2}$, i.e., either a round two-sphere or one of the Borůka-Veronese 2-spheres in $S^{2m}$. This entails a classification of all Willmore $\mathbb{C} P^1$ in $S^{2m}$. As a second main result we show that there exists no homogeneous Willmore upper-half plane in $S^{n+2}$ and we give, in terms of special constant potentials, a simple loop group characterization of all homogeneous surfaces which have an abelian transitive group.