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.
New criteria found for Willmore submanifolds in Lie group orbits.
problem Criteria for Willmore submanifolds in Lie group orbits.
method Criteria for Willmore submanifolds based on orbit type stratification.
result Found Willmore orbits in each stratified subset of orbit type.
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.
Researchers identify only two types of tori with specific energy constraints.
problem Finding constrained Willmore tori in 3-space with specific energy limits.
method Analyzing isothermic constrained Willmore tori in the 3-sphere.
result Homogeneous and 2-lobe Delaunay tori are the only isothermic constrained Willmore tori with Willmore energy below 8π.
The paper finds new constrained Willmore minimizers for non-rectangular tori.
problem Finding constrained Willmore minimizers for non-rectangular tori.
method Analyzing immersed tori in 3-space to minimize Willmore energy.
result The candidates constructed in previous work are constrained Willmore minimizers in certain non-rectangular conformal classes.
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…
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…
Researchers create families of tori minimizing Willmore energy.
problem Finding minimizers of Willmore energy for non-rectangular tori.
method Explicit construction of 1D families of embedded constrained Willmore tori.
result Candidates for minimizers are explicitly constructed and shown to minimize Willmore energy.
The Willmore energy for Frenet curves in quaternionic projective space is the generalization of the Willmore functional for immersions into the 4-sphere. Critical points of the Willmore energy are called Willmore curves in quaternionic projective space. Using a Baecklund transformation on Willmore curves, we generalize…
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. 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.
Paper proves convergence for Willmore immersions with minimal bubbles.
problem Proving convergence of Willmore immersions with minimal bubbles.
method Replaces total curvature control with local Willmore energy control.
result Proves convergence result for sequences of Willmore immersions.
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.
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.
New proof of Willmore conjecture using tori minimizers.
problem Proving the Willmore conjecture in 3-space.
method Minimizing the Willmore energy of tori in S3. result 2-lobed Delaunay tori uniquely minimize Willmore energy.
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.
Rigidity for 4D Willmore submanifolds with boundary.
problem Understanding critical points of Willmore energy with boundary conditions.
method Proving a 4-Willmore equation and establishing curvature estimates.
result Four dimensional Willmore submanifolds with totally geodesic boundary are umbilic.
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.
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.
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}…
Using the reformulation in divergence form of the Euler-Lagrange equation for the Willmore functional as it was developed in "Analysis of the Willmore Functional" by T. Riviere (Invent. Math. 174), we study the limit of a local Palais-Smale sequence of weak Willmore immersions with locally square-integrable second fund…
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…
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 …
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.
Study of tori of revolution under Willmore flow converges to Clifford Torus.
problem Long-time behavior and convergence of Willmore flow for tori of revolution.
method Gradient flow of Willmore energy for tori of revolution, analyzing energy threshold and convergence to Clifford Torus.
result Convergence of Willmore flow to Clifford Torus for initial energy below 8π.
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 study bounds Morse indices of Willmore spheres in relation to min-max sweep-outs.
problem Estimating Morse indices of Willmore spheres.
method Analyzing the sum of Morse indices of Willmore spheres in min-max sweep-outs.
result At most one Willmore sphere can have index 1 among those realising min-max sphere eversion.
In this paper we classify branched Willmore spheres with at most three branch points (including multiplicity), showing that they may be obtained from complete minimal surfaces in R3 with ends of multiplicity at most three. This extends the classification result of Bryant. We then show that this may be applied to …
The paper studies spheres with small diameter in 3D manifolds concentrating at scalar curvature critical points.
problem Understanding the behavior of Willmore spheres with small diameter in 3D manifolds.
method Analyzes spheres under bounded Willmore energy and small diameter constraints, focusing on scalar curvature critical points.
result Embedded Willmore spheres concentrate at critical points of scalar curvature under small diameter and bounded energy conditions.
Study of Willmore energy on sphere sublevel sets and flow singularities.
problem Understanding the Willmore energy landscape and singularities of the Willmore flow.
method Gluing different instances of the Willmore flow and using an invariant for triple-point-free spheres.
result Classification of initial surfaces with energy at most 12π leading to unavoidable singularities.
Study on stability of free boundary Willmore problem using new gradient inequality.
problem Stability of free boundary Willmore problem.
method New Łojasiewicz-Simon gradient inequality for functionals on infinite dimensional manifolds.
result Existence and convergence of solutions for the free boundary Willmore flow.
The paper proves inequalities for hypersurfaces in weighted manifolds.
problem Willmore-type inequalities for closed hypersurfaces in weighted manifolds.
method Analyzes weighted manifolds with nonnegative Bakry-Émery Ricci curvature, proving sharp inequalities and characterizing equality cases.
result Derives sharp Willmore-type and Willmore-like inequalities in steady and shrinking gradient Ricci solitons.
Quantizes Willmore energy in Riemannian manifolds with bounded energy and area.
problem Quantization of Willmore energy in bounded energy and area conditions.
method Uniform boundedness of Willmore energy and area, weak convergence of maps, and conformal structures in compact domain.
result Quantization of Willmore energy holds under specified conditions.
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.
The study classifies branched Willmore spheres in 3- and 4-spheres.
problem Classifying branched Willmore spheres in 3- and 4-spheres.
method Analyzing variational branched Willmore spheres and their projections onto R3 and R4. result Improved C1,1 regularity of unit normals and integer multiples of width in sphere eversion. Starting from suitable tableaux over finite dimensional Lie algebras, we provide a scheme for producing involutive linear Pfaffian systems related to various classes of submanifolds in homogeneous spaces which constitute integrable systems. These include isothermic surfaces, Willmore surfaces, and other classical solit…
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…
Let x:M→Sn+p be an n-dimensional submanifold in an (n+p)-dimensional unit sphere Sn+p, x:M→Sn+p is called a Willmore submanifold to the following Willmore functional: ∫M(S−nH2)2ndv, where S=α,i,j∑(hijα)2 is the square of the length of the second fundam…
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.
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.
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. 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.
Ejiri's torus in S5 is the first example of Willmore surface which is not conformally equivalent to any minimal surface in any space forms. Li and Vrancken classified all Willmore surfaces of tensor product in Sn by reducing them into elastic curves in S3, and the Ejiri torus appeared as a special example. I…
The paper proves global existence and convergence of Möbius-invariant Willmore flow in 3-sphere.
problem Global existence and convergence of Möbius-invariant Willmore flow in 3-sphere.
method Use of invariant center manifolds and recent achievements about the Möbius-invariant Willmore flow.
result Fully and smoothly convergent flow lines are stable w.r.t. small perturbations.
The paper studies Willmore surfaces in 4D conformal manifolds and finds the Clifford torus is strictly Willmore-stable.
problem Exploring the Willmore functional for surfaces in 4D conformal manifolds.
method Detailed calculation of first and second variations, derivation of Euler-Lagrange equation in a conformally invariant form.
result The Clifford torus in CP2 is strictly Willmore-stable, supporting a conjecture. 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. 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…