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. 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.
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.
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.
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. 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. 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.
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.
Existence of minimizers for Klein bottles in 4D space.
problem Finding Klein bottles with minimal Willmore energy in 4D space.
method Analyzing immersed Klein bottles in Euclidean 4-space, proving existence and properties of minimizers.
result Existence of infinitely many distinct embedded Klein bottles with specific properties.
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. 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…
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.
Study on Willmore spheres, calculating their index and relating it to minimal surfaces.
problem Calculating the index of Willmore spheres and understanding its relation to minimal surfaces.
method Analyzing inverted complete minimal surfaces with embedded planar ends, computing Morse Index, and relating it to Jacobi fields.
result Computed the index of a Willmore sphere as \(m-d\), where \(m\) is the number of ends and \(d\) is the dimension of normals at the \(m\)-fold point.
Minimal existence time for Willmore flow established for smooth and weak Lipschitz initial data.
problem Existence time of the Willmore flow for various initial conditions.
method Established minimal existence time for Willmore flow using geometric data and conservation laws.
result Minimal existence time is a function of geometric data for general weak Lipschitz initial data.
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.
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.
Study minimizes Willmore energy with constraints on surface properties.
problem Minimizing Willmore energy under specific surface properties.
method Adapting Keller-Mondino-Rivière, Bauer-Kuwert, and Ndiaye-Schätzle methods.
result Existence of smooth minimizers for a broad range of constraints.
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. Proves strict inequality for minimizers of Willmore energy under isoperimetric constraints.
problem Minimizing the Willmore energy under isoperimetric constraints.
method Connected sum approach, building on previous work by Keller-Mondino-Rivière.
result Existence of minimizers for the isoperimetric constrained Willmore problem in every genus.
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 classify complete surfaces of Willmore type with positive mean curvature in Riemannian three-manifolds.
Removes singularity order for Willmore immersions, reducing bubbling scenarios.
problem Understanding the singularity order of weak limits of Willmore immersions.
method Obtains removability result on singularity order, reducing bubbling scenarios.
result Only three out of twelve non-planar minimal surfaces may occur as bubbles of Willmore immersions.
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.
Motivated by a simple model for elastic cell membranes, we minimize the Willmore functional among two-dimensional spheres embedded in R^3 with prescribed isoperimetric ratio.
Method calculates Morse index of branched Willmore spheres in 3-space.
problem Computing the Morse index of branched Willmore spheres.
method Developed a method to compute the Morse index using a matrix whose dimension is equal to the number of ends of the dual minimal surface.
result Found that for all immersed Willmore spheres, the Morse index is less than or equal to the number of ends minus one.
In this work we present new fundamental tools for studying the variations of the Willmore functional of immersed surfaces into Rm. This approach gives for instance a new proof of the existence of a Willmore minimizing embedding of an arbitrary closed surface in arbitrary codimension. We explain how the same approach…
Study on surfaces minimizing Willmore functional with small isoperimetric ratio converging to zero.
problem Analyzing surfaces in R3 that minimize the Willmore functional with a small isoperimetric ratio. method Full blowup analysis of the limit when the isoperimetric ratio converges to zero.
result Two spheres connected by a catenoidal neck in the limit.
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.
We develop a bubble tree construction and prove compactness results for W2,2 branched conformal immersions of closed Riemann surfaces, with varying conformal structures whose limit may degenerate, in a compact Riemannian manifold with uniformly bounded areas and Willmore energies. The compactness property is appli…
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 prove a Willmore conjecture for surfaces with specific symmetries.
problem Finding the surface in S3 that minimizes the Willmore energy with given topological type. method Local computation of the orbifold Euler number to exclude certain intersection patterns of surfaces with symmetries.
result Lawson's minimal surface ξg,1⊂S3 minimizes the Willmore energy among surfaces of genus g>1 with the same symmetries. 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 …
Proves Willmore conjecture for surfaces with specific symmetries.
problem Finding surfaces in S3 that minimize Willmore energy. method Analyzes symmetries and uses group theory to prove minimization.
result Lawson's minimal surfaces minimize Willmore energy under their symmetries.
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…
Study of immersions with Willmore energy leading to spherical and catenoid bubbles.
problem Classifying immersions with specific energy properties.
method Analyzing sequences of weak immersions with diverging conformal classes, applying Möbius transformations, and strong Wloc2,2-limits. result Obtaining spherical and catenoid bubbles as limits of immersions.
Study of higher-dimensional Willmore energies via minimal submanifold asymptotics.
problem Understanding conformally invariant generalizations of Willmore energy.
method Derives and studies a new energy functional for submanifolds, connects it to minimal submanifold asymptotics in Poincare-Einstein spaces.
result Explicitly identifies the energy for four-dimensional submanifolds and studies its variational properties.
The family of Willmore immersions from a Riemann surface into Sn+2 can be divided naturally into the subfamily of Willmore surfaces conformally equivalent to a minimal surface in Rn+2 and those which are not conformally equivalent to a minimal surface in Rn+2. On the level of their conformal Gauss maps…
Study on smoothness of 4D Willmore-type hypersurfaces.
problem Investigating smoothness of critical points of a 4D Willmore-type energy.
method Computed first variation, applied Noether's theorem, investigated other generalizations.
result Critical points of the energy are smooth.
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.
Investigates a new four-dimensional energy related to Willmore energy.
problem Exploring a new conformally invariant energy in four dimensions.
method Computed first variation, applied Noether's theorem, investigated other generalizations.
result Critical points of the new energy are smooth and do not include minimal hypersurfaces.
New formulas for minimal surfaces with specific end conditions.
problem Existence and explicit formulas for minimal surfaces with embedded planar ends.
method Provided new explicit formulas for genus 0 minimal surfaces in R^3 with 2k+1 embedded planar ends.
result Existence and explicit formulas for minimal surfaces with 2k+1 embedded planar ends for all k ≥ 4.
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…
Study of ends of immersed minimal and Willmore surfaces in asymptotically flat spaces.
problem Understanding the behavior of ends of immersed surfaces in asymptotically flat spaces.
method Analyzing the asymptotic behavior of immersed surfaces with bounded second fundamental form in spaces with specific decay properties.
result Precise asymptotic behavior of ends of immersed surfaces is determined, depending on the ambient metric decay.
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. 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.
The Clifford torus is unique when its isoperimetric ratio is prescribed.
problem Proving the uniqueness of the Clifford torus with a prescribed isoperimetric ratio.
method Reduction to a positivity question of a polynomial recurrence.
result The conjecture can be reduced to a polynomial recurrence positivity question.
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.
Connected surfaces with boundary minimize Willmore energy under certain conditions.
problem Finding connected compact surfaces with minimal Willmore energy.
method Minimizing the Willmore energy on integer rectifiable curvature varifolds with boundary constraints.
result Existence of connected minimizers when the infimum of the problem is less than 4π.