The goal of the present paper is to investigate the algebraic structure of global conformal invariants of submanifolds. These are defined to be conformally invariant integrals of geometric scalars of the tangent and normal bundle. A famous example of a global conformal invariant is the Willmore energy of a surface. In …
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48 results for “Willmore integrand”
Study proves uniform regularity for surface energies, critical and subcritical.
problem Establishing regularity for surface energies in critical and subcritical cases.
method Uniform ε-regularity estimates for intrinsic elliptic Lagrangians.
result Critical points of surface energies are uniformly regular for a wide class of Lagrangians.
New geometric interpretation of discrete Willmore energy using rolling spheres connection.
problem Discrete formulation of Willmore energy for simplicial surfaces.
method Geometric interpretation of Möbius invariant discrete Willmore energy using rolling spheres connection.
result Clear geometric interpretations of discrete Willmore energy with manifest Möbius invariance.
For two-dimensional, immersed closed surfaces , we study the curvature functionals and with integrands and , respectively. Here is the second fundamental form, is the mean curvature and we assume . Our main result asser…
Chiral string integrands simplify to ambitwistor string integrands in the tensionless limit.
problem Understanding the relationship between chiral and ambitwistor string integrands.
method Analyzing the tensionless limit of chiral superstring integrands.
result Chiral superstring integrands reduce to ambitwistor string integrands in the tensionless limit.
This paper proves properties of convex integrands and their duals.
problem Properties of convex integrands and their duals.
method Analyzing and stable convex integrands.
result Dual convex integrands of stable convex ones are also stable.
Stable convex integrands are open and dense in smooth convex integrands.
problem Characterizing stability of convex integrands.
method Whitney topology analysis.
result Set of stable convex integrands is open and dense.
Simultaneous smoothness and simultaneous stability of a strictly convex integrand and its dualmath.GT
The paper examines smoothness and stability of convex integrands and their duals.
problem Properties of convex integrands and their duals.
method Investigation of simultaneous smoothness and stability of a strictly convex integrand and its dual.
result For a strictly convex integrand, its dual is of class if and only if the integrand is strictly convex.
Kernel quadratures can be consistent even when the integrand is less smooth than assumed.
problem Kernel quadratures assume smoothness of integrands, but this assumption is often violated in practice.
method Derives convergence rates for kernel quadratures in misspecified settings, relating them to the lesser smoothness of the integrand.
result Kernel quadratures can be consistent even when the integrand is less smooth than assumed, providing alternatives to Monte Carlo integration.
Quantum speedup for Monte Carlo integration reduces integrand calls.
problem Reducing the number of calls to the integrand subroutine in high-dimensional Monte Carlo integration.
method Combining nested quantum amplitude estimation with pseudorandom numbers for separable integrands.
result Significant reduction in the number of integrand calls for high-dimensional integration.
Ambitwistor string matches superstring chiral integrands at zero tension.
problem Matching scattering amplitudes in superstring theory and ambitwistor string theory.
method Direct computation and reduction to ordinary moduli space.
result Chiral half integrands of superstring match those of ambitwistor string in the zero tension limit.
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…
In anomaly-free quantum field theories the integrand in the bosonic functional integral--the exponential of the effective action after integrating out fermions--is often defined only up to a phase without an additional choice. We term this choice ``setting the quantum integrand''. In the low-energy approximation to M-t…
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 .
method Explicit construction of minimal bubbles and analysis of limit sequences of Willmore immersions.
result Compactness for immersed Willmore tori of energy below 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.
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.
New geometric interpretations reveal structure of AC integrands.
problem Understanding and verifying the atomic condition for integrands.
method Reinterpretation of atomic condition in convex geometry.
result Quantitative versions EC and QEC proposed; stability and regularity results.
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.
Developed a method to compute the cost of sphere eversion using Willmore energies.
problem Computing the cost of sphere eversion in 3D Euclidean space.
method Minmax procedure for constructing Willmore surfaces and applying it to sphere eversion.
result Computed the cost of sphere eversion in terms of Willmore energies.
In this paper, it is shown that a Wulff shape is strictly convex if and only if its convex integrand is of class . Moreover, applications of this result are given.
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 .
result 2-lobed Delaunay tori uniquely minimize Willmore energy.
Bayesian optimization for expensive integrands achieves optimal performance.
problem Optimizing functions with expensive integrands in noisy conditions.
method Bayesian optimization with discretization-free value of information optimization.
result Achieves optimal performance in noisy and smooth conditions.
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.
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.
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.
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 .
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.
Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms, a topic in Lorentzian conformal geometry which parallels the theory of Willmore surfaces in , 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…
Willmore surfaces in spheres via loop groups IV: on totally isotropic Willmore two-spheres in math.DG
Totally isotropic surfaces in 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 . 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.
Extends Euler calculus to continuous integrands using curvature.
problem Limitations of Euler calculus with simple functions.
method Integrates with respect to Gaussian curvature within O-minimal theories.
result Satisfies a Fubini theorem and extends to a functor.
Branched Willmore Spheresmath.DG
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 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 and .
result Improved regularity of unit normals and integer multiples of width in sphere eversion.
The classification of Willmore 2-spheres in the -dimensional sphere is a long-standing problem, solved only when by Bryant, Ejiri, Musso and Montiel independently. In this paper we give a classification when . There are three types of such surfaces up to Möbius transformations: (1) super-conformal…