The paper classifies Willmore Legendrian surfaces in S^5 and studies their properties.
problem Classifying and understanding Willmore Legendrian surfaces in S^5.
method Using an equality from Luo's work, the authors relate Willmore Legendrian surfaces to contact stationary Legendrian surfaces and prove classification results.
result Classification of Willmore Legendrian spheres in S^5 and integral inequality for Willmore Legendrian surfaces.
Minimal Legendrian surfaces found in 5D sphere.
problem Characterizing Willmore Legendrian surfaces in S5. method Analyzing properties of Willmore and csL Willmore surfaces.
result Complete Willmore Legendrian surfaces in S5 are minimal. Let $ X: M \hook S^5$ be a compact Legendrian surface in pseudoconformal(CR) 5-sphere. We introduce a pseudoconformally invariant Willmore type second order functional $ \W(X)$, and study its critical points called Willmore Legendrian surfaces. The fifth order structure equations show that Willmore dual can be defined …
Formula proves almost monotonicity for H-minimal surfaces in Heisenberg group.
problem Analyzing H-minimal Legendrian surfaces in Heisenberg group.
method Proved an almost monotonicity formula.
result Deduced a Bernstein-Liouville type theorem.
Study CR-geometry analog of conformal volume for spheres' submanifolds.
problem Analog of Li-Yau conformal volume in CR-geometry.
method Associate invariant quantity to submanifolds of spheres.
result Introduced CR-Volume for horizontal submanifolds of spheres.
Study fourth-order geometric problems on Willmore surfaces.
problem Fourth-order geometric problems on Willmore surfaces.
method Local energy estimates and global gap lemma derivation.
result Proved several local energy estimates and derived a global gap lemma.
Willmore flow preserves low energy surfaces to planes.
problem Preserving low energy surfaces to planes under Willmore flow.
method Willmore flow equation for complete, properly immersed surfaces in Rn.
result Complete Willmore surfaces with low energy converge to planes.
The paper disproves compactness for high-energy Willmore immersions and finds minimal bubbles on Willmore surfaces.
problem Compactness for high-energy Willmore immersions of Willmore energy above 16π. method Explicit construction of minimal bubbles and analysis of limit sequences of Willmore immersions.
result Compactness for immersed Willmore tori of energy below 12π is proven. Survey of Willmore surfaces in spheres using DPW method.
problem Global and local properties of Willmore surfaces in spheres.
method DPW method for conformal Gauss map.
result Characterizations of minimal surfaces and Willmore deformations.
The paper describes a new method for Willmore surfaces in spheres.
problem Finding new examples of Willmore surfaces in spheres.
method DPW approach via conformal Gauss maps.
result New examples of Willmore surfaces, including a two-sphere in S6. Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms, a topic in Lorentzian conformal geometry which parallels the theory of Willmore surfaces in S4, are studied in this paper. We define two kinds of transforms for such a surface, which produce the so-called left/right polar surfaces and the adjoint s…
The paper discusses Gauss maps for Möbius surfaces in spheres and their applications to Willmore surfaces.
problem Understanding Gauss maps and their relation to Willmore surfaces in spheres.
method Definition and study of Lorentzian 2-plane lifts for Möbius surfaces, and equivalence to Willmore condition.
result The conformal harmonicity of a Lorentzian 2-plane lift is equivalent to the Willmore condition for a surface.
Classifies surfaces with no Gaussian curvature.
problem Classifying surfaces with vanishing Gaussian curvature.
method Analyzes Willmore surfaces, studies Willmore cones, gives a Bernstein-type theorem.
result Classifies simply-connected, complete Willmore surfaces with vanishing Gaussian curvature.
Removability result for Willmore surfaces in arbitrary codimension.
problem Removability of singularities in Willmore surfaces.
method Analyzing Willmore surfaces and their removability in arbitrary codimension.
result Only three families of non-planar minimal surfaces can occur in Willmore min-max problems.
Exploring conjectures in constrained Willmore problem.
problem Understanding the Willmore functional over compact surfaces.
method Analyzing conjectures from partial results and numerical experiments.
result Ramifications for deeper understanding of the Willmore functional.
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 complex projective space or the inversion of a minimal…
Totally isotropic surfaces in S6 are not necessarily Willmore surfaces. Therefore it is the first goal of this paper to derive a geometric characterization of totally isotropic Willmore two-spheres in S6. This will naturally yield to a description of such surfaces in terms of the loop group language. Moreover, ap…
The classification of Willmore 2-spheres in the n-dimensional sphere Sn is a long-standing problem, solved only when n=3,4 by Bryant, Ejiri, Musso and Montiel independently. In this paper we give a classification when n=5. There are three types of such surfaces up to Möbius transformations: (1) super-conformal…
We view conformal surfaces in the 4--sphere as quaternionic holomorphic curves in quaternionic projective space. By constructing enveloping and osculating curves, we obtain new holomorphic curves in quaternionic projective space and thus new conformal surfaces. Applying these constructions to Willmore surfaces, we show…
New symmetric Willmore tori emerge from Clifford torus in Berger spheres.
problem Finding new symmetric Willmore surfaces from Clifford torus.
method Applying bifurcation theory to estimate Morse index of Willmore surfaces.
result New symmetric Willmore tori emerge from Clifford torus.
Two classification theorems for Willmore surfaces in S² × S².
problem Classifying Willmore surfaces in S² × S².
method Analytical proofs for minimal and product type surfaces.
result Classification of Willmore surfaces in S² × S².
The paper finds formulas for Willmore surfaces and discusses symmetry breaking.
problem Understanding symmetry and symmetry breaking in Willmore surfaces.
method Proved explicit formulas and demonstrated symmetry breaking examples.
result Symmetric boundary conditions do not guarantee symmetric surfaces.
A neural flow method minimizes Willmore energy for 2-surfaces in 3D space.
problem Minimizing Willmore energy for closed oriented 2-surfaces in 3D space.
method Introducing neural Willmore flow to model and minimize the Willmore energy using neural architectures.
result The neural flow reproduces expected round sphere and Clifford torus for genus 0 and 1 surfaces, respectively, and finds minimal Willmore surfaces for genus 2.
The Willmore flow preserves surface volume, leading to convergence to a sphere.
problem Long-term behavior of volume-preserving Willmore flow on surfaces.
method Volume-preserving Willmore flow, blow-up analysis, constrained Lojasiewicz-Simon inequality.
result Smooth solutions exist for spherical surfaces with Willmore energy below 8π and converge to a sphere.
Energy quantization for surfaces with area, volume, and mean curvature constraints.
problem Energy quantization for constrained Willmore surfaces.
method Established through strong compactness under energy thresholds.
result Strong compactness of constrained Willmore surfaces, including minimizers.
Classification of special surfaces in spheres.
problem Classifying homogeneous Willmore surfaces in spheres.
method Analyzing properties of homogeneous Willmore surfaces and their conformal equivalences.
result Classification of all Willmore surfaces in specific spheres.
The paper shows deformations between minimal surfaces in Sn+2 and Hn+2.
problem Deformation of minimal surfaces between Sn+2 and Hn+2. method Willmore deformation approach.
result Existence of smooth families of Willmore surfaces connecting minimal surfaces in Sn+2 and Hn+2. In this note we demonstrate how the analogy between the harmonic Gauss map of a constant mean curvature surface and the harmonic conformal Gauss map of a Willmore surface can be used to obtain results on Willmore surfaces.
The study constructs equivariant harmonic maps into symmetric spaces with applications to Willmore surfaces.
problem Constructing harmonic maps into symmetric spaces.
method Equivariant primitive harmonic maps construction.
result Examples of S1-equivariant Willmore Moebius strips in S3. This paper characterizes Willmore surfaces in spheres using harmonic maps.
problem Characterizing Willmore surfaces in spheres using harmonic maps.
method Analyzes the conformal Gauss map of Willmore surfaces and its properties.
result Generically, strongly conformally harmonic maps from Riemann surfaces to the target space are associated with Willmore surfaces in spheres.
We develop a general Minmax procedure in Euclidian spaces for constructing Willmore surfaces of non zero indices. We implement this procedure to the Willmore Minmax Sphere Eversion in the 3 dimensional euclidian space. We compute the cost of the Sphere eversion in terms of Willmore energies of Willmore Spheres in ${\R}…
Smooth minimizers found for Willmore energy surfaces.
problem Finding minimizers for Willmore energy surfaces.
method Existence and smoothness established through axially symmetric surfaces with prescribed isoperimetric ratio.
result Existence and smoothness of minimizers proven.
The paper proves topological finiteness for surfaces with finite Willmore energy.
problem Understanding the topology of surfaces with finite Willmore energy.
method Combining Allard regularity theorem and Reifenberg's topological disk theorem.
result Topological finiteness for a class of properly immersed surfaces with finite Willmore energy.
Study of conformal Gauss map for Willmore surfaces in model spaces.
problem Characterizing Willmore surfaces and their geometric properties.
method Detailed study of conformal Gauss map and its application to Willmore surfaces.
result New characterizations of minimal and CMC surfaces using conformal Gauss map.
Researchers define and prove existence of minimizers for generalized Willmore functionals.
problem Existence of area constrained minimizers for generalized Willmore functionals.
method Compactness result for branched, immersed, stratified surfaces; direct minimization; introduction of haunted surfaces.
result Existence of area constrained minimizers for generalized Willmore functionals.
Study of umbilic points on Willmore surfaces in 3-sphere.
problem Characterizing umbilic points on Willmore surfaces.
method Analysis of conformal Gauss map and Gauss-Bonnet formula.
result Unified expression for Willmore energy in space-forms.
Willmore surfaces have umbilic points forming a smooth manifold.
problem Characterizing umbilic points on Willmore surfaces.
method Analytic proof of the umbilic set's structure.
result Umbilic set forms a smooth manifold.
We discuss several kinds of Willmore surfaces of flat normal bundle in this paper. First we show that every S-Willmore surface with flat normal bundle in Sn must locate in some S3⊂Sn, from which we characterize Clifford torus as the only non-equatorial homogeneous minimal surface in Sn with flat normal…
In this paper we provide a systematic discussion of how to incorporate orientation preserving symmetries into the treatment of Willmore surfaces via the loop group method. In this context we first develop a general treatment of Willmore surfaces admitting orientation preserving symmetries, and then show how to induce f…
The study classifies surfaces in Berger spheres as Willmore and Hopf tori.
problem Classifying surfaces in Berger spheres as Willmore and Hopf tori.
method Defined a Willmore functional for surfaces in homogeneous spaces and computed its variational formula. Characterized Clifford and Hopf tori as Willmore surfaces satisfying a sharp inequality.
result Clifford and Hopf tori are the only Willmore surfaces in Berger spheres satisfying a specific inequality.
Proves conditions for Willmore surfaces to have finite ends or finite total curvature.
problem Conditions for Willmore surfaces to have finite ends or finite total curvature.
method Analyzes scale-invariant second fundamental form near infinity.
result Proves conditions for Willmore surfaces to have finite ends or finite total curvature.
Local foliation of 3D manifolds by Willmore surfaces with curvature constraints.
problem Constructing foliations of manifolds by surfaces of Willmore type.
method Adapting a method from constant mean curvature foliations to Willmore surfaces.
result Existence of a local foliation of a 3D Riemannian manifold by Willmore critical points with area constraint.
We found a new formulation to the Euler-Lagrange equation of the Willmore functional for immersed surfaces in Rm. This new formulation of Willmore equation appears to be of divergence form, moreover, the non-linearities are made of jacobians. Additionally to that, if $\bH$ denotes the mean curvature vector of the…
A new approach is proposed for study structure and properties of the total squared mean curvature W of surfaces in R3. It is based on the generalized Weierstrass formulae for inducing surfaces. The quantity W (Willmore functional) is shown to be invariant under the modified Novikov--Veselov hierarchy of in…
Study compact Willmore surfaces without complex structure convergence, computing energy loss and geodesic lengths.
problem Compactness of Willmore surfaces without complex structure convergence.
method Compute energy loss in neck and geodesic lengths in Grassmannian G(2,n). result Limit of Gauss map image is a geodesic in G(2,n) with computable length. Constrained Willmore surfaces are conformal immersions of Riemann surfaces that are critical points of the Willmore energy W=∫H2 under compactly supported infinitesimal conformal variations. Examples include all constant mean curvature surfaces in space forms. In this paper we investigate more generally the crit…
After the surface theory of Möbius geometry, this study concerns a pair of conformally immersed surfaces in n-sphere. Two new invariants θ and ρ associated with them are introduced as well as the notion of touch and co-touch. This approach is helpful in research about transforms of certain surface classes. As an …
This is the first comprehensive introduction to the authors' recent attempts toward a better understanding of the global concepts behind spinor representations of surfaces in 3-space. The important new aspect is a quaternionic-valued function theory, whose "meromorphic functions" are conformal maps into quaternions, wh…