In this paper we consider two special classes of constrained Willmore tori in the 3-sphere. The first class is given by the rotation of closed elastic curves in the upper half plane - viewed as the hyperbolic plane - around the x-axis. The second is given as the preimage of closed constrained elastic curves, i.e., elas…
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48 results for “constrained elastic curves”
New discrete curves defined in space forms with geometric properties.
problem Defining discrete elastic and constrained elastic curves in space forms.
method Extending discrete Euclidean curvature to space forms and using Bäcklund transformations.
result Discrete elastic and constrained elastic curves are elements of a curve hierarchy.
Study geodesics in constrained curve spaces, including elastic curves and concentric circles.
problem Geodesics in constrained curve spaces with Sobolev metrics.
method Intrinsic and constructive approaches.
result Construct geodesics in elastic curve and concentric circle spaces.
The paper studies surfaces with spherical curvature lines and their generation by constrained elastic curves.
problem Understanding surfaces with spherical curvature lines and their generation mechanisms.
method The approach involves Lie sphere transformations, Legendre curves, and polynomial conserved quantities of connections.
result Lie applicable surfaces with exactly one family of spherical curvature lines are generated by the lift of constrained elastic curves.
Study curves evolving by gradient flow of elastic energy, proving existence, smoothing, and convergence.
problem Evolution of curves with fixed length and clamped boundary conditions.
method Negative -gradient flow of elastic energy, existence, parabolic smoothing, constrained Lojasiewicz-Simon gradient inequality.
result Convergence to a critical point as time tends to infinity.
Study curves evolving on hypersurfaces with free boundaries, preserving length.
problem Evolution of curves on hypersurfaces with free boundaries.
method Nonlocal evolution equation with nonlinear boundary conditions, short-time existence, uniqueness, and parabolic energy estimates.
result Global existence and convergence to critical points proved.
The paper studies how curves evolve under area constraints and converges to a critical point.
problem Evolution of plane curves with fixed area under elastic energy gradient.
method Local and global existence of the flow, simplicity assumption, Łojasiewicz--Simon inequality.
result The evolving curve's length remains bounded and converges to a critical point.
Physics-constrained GP predicts material states under shockwave conditions.
problem Predicting material states under extreme shockwave conditions.
method Physics-constrained Gaussian Process regression with Rankine-Hugoniot constraints.
result Reproduces Hugoniot curves with satisfactory accuracy and uncertainty quantification.
Transforms curves and surfaces for efficient geometric analysis.
problem Efficiently analyzing and comparing curves and surfaces.
method Square root velocity transformation for curves and intrinsic comparison for surfaces.
result Fundamental geometric properties of curves under the transformation.
Approximate 3D elastic curves with exact constraints
problem Designing and approximating 3D elastic curves
method Numerically stable method for recovering 11 parameters
result Fast and stable approximation of arbitrary curves
Study preserves planar and graphical properties of curves under elastic flow.
problem Maintaining planar and graphical properties of non-compact curves under elastic flow.
method Extended recent work on adapted elastic energy to derive thresholds for planar and graphical embeddedness.
result Derived new Li--Yau type inequality for complete planar curves.
New insights into stability of special curves on spheres.
problem Stability of closed -elastic curves on spheres.
method Analytical proof and construction of curves.
result All closed spherical -elastic curves for are unstable.
Migrating elastic flowsmath.AP
Study on migrating elastic flows of curves across half-planes.
problem Migrating elastic flows of curves from upper to lower half-planes.
method Analytical and numerical construction of migrating elastic flows.
result Construction of various migrating elastic flows.
Study on elastic curves with variable stiffness, derived from bending energy.
problem Modeling elastic wires with varying thickness.
method Derive Euler-Lagrange equations for curves with variable bending stiffness.
result Characterizations of elastic curves with variable stiffness.
Paper tackles shape graph registration using neural networks.
problem Constrained registration of shape graphs with varying nodes and edges.
method Shape-Graph Matching Network (SGM-net) with an elastic shape metric loss function.
result State-of-the-art matching performance and reduced computational cost.
The paper studies the free elastic flow of closed curves and finds their asymptotic shape converges to a circle.
problem Challenges in studying the asymptotic behavior of the free elastic flow for closed curves.
method Analysis of the free elastic flow as an -gradient flow for Euler's elastic energy.
result An appropriate rescaling of initial curves geometrically close to circles converges to a unique round circle.
Migrating elastic flows IImath.AP
Solves curve migration problem with elastic flows.
problem Curve migration problem with natural boundary conditions.
method Constructing migrating elastic flows.
result Extends previous work to purely local flow.
Study on closed -elastic curves in hyperbolic and de Sitter planes.
problem Existence of closed -elastic curves with nonconstant curvature.
method Analysis of -elastic curves in hyperbolic and de Sitter planes.
result Existence of closed -elastic curves in hyperbolic plane for ; in de Sitter plane for .
Ejiri's torus in 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 by reducing them into elastic curves in , and the Ejiri torus appeared as a special example. I…
Unified survey of elastic flow for curves and networks.
problem Understanding the evolution of curves and networks under elastic forces.
method Unified presentation and proof of global existence and convergence for closed curves.
result Global existence and smooth convergence to critical points for closed curves in R^2.
The paper finds curves minimizing elastic energy pinned at endpoints.
problem Finding curves that minimize elastic energy with fixed endpoints.
method Applying the shooting method to identify and classify critical points.
result Critical points consist of wavelike elasticae, and minimizers have no loops or interior inflection points.
Optimal thresholds ensure curves remain embedded in flows.
problem Preserving the embeddedness of elastic flows of curves.
method Variational characterization and minimization of bending energy.
result Optimal thresholds for preserving embeddedness are found.
Scheme minimizes -elastic energy of curves over time.
problem Minimizing -elastic energy of curves over time.
method Minimizing movement scheme with approximate normal graphs.
result Short-time existence and lower bound on solution's lifetime.
New curves defined by curvature powers studied for variational properties.
problem Characterizing translating solitons in curve flows.
method Variational characterization of generalized elastic curves.
result New variational characterization of grim reaper curve.
The study examines elastic curves with self-intersections and their properties.
problem Understanding self-intersecting elastic curves in Euclidean space.
method Review of classical elastica theory and discussion of open problems.
result Proof of Li--Yau type inequality for self-intersecting curves.
Study calculates the elastic energy of curves on a sphere.
problem Elastic energy of curves on a sphere.
method Introduced -curvature functional for rectifiable curves in the sphere and proved its finiteness.
result The -curvature functional agrees with the integral of geodesic curvature raised to the power for curves in .
The energy minimization problem associated to uniform, isotropic, linearly elastic rods leads to a geometric variational problem for the rod centerline, whose solutions include closed, knotted curves. We give a complete description of the space of closed and quasiperiodic solutions. The quasiperiodic curves are paramet…
Flow deforms locally convex curves into target curves.
problem Deforming locally convex curves to target curves with same elastic energy.
method Curvature flow with nonlocal term to evolve curves.
result Flow deforms curves to target curves if elastic energies match.
Study on elastic curves pinned at the boundary, focusing on minimizers and their interaction with obstacles.
problem Minimizing elastic bending energy for open planar curves with obstacles.
method Investigation of global minimizers and explicit solutions for different values of the penalization parameter.
result Explicit threshold for above which minimizers touch the obstacle, regardless of obstacle shape.
The elastic flow of curves converges smoothly to a critical point.
problem Smooth convergence of elastic flow of curves.
method Application of Lojasiewicz-Simon inequality.
result Smooth convergence to a critical point.
Study on dynamic curves with elastic energy and spontaneous curvature.
problem Modeling and analyzing dynamic planar curves with elastic energy.
method Gradient flow of inclination angle, nonlocal quasilinear system, local well-posedness, global existence, convergence.
result Local well-posedness, global existence, convergence of the flow for weak regularity initial data.
Study on hyperbolic elastic flow, proving convergence and quantifying singularities.
problem Understanding singularities and convergence of elastic flow in hyperbolic plane.
method Analyzes closed and open curves with clamped boundary conditions, proving convergence without small energy assumption.
result Each singularity carries an energy cost of at least 8, and blow-ups are explicitly classified.
In this paper, we consider the classical variational problem in the Galilean space. we develop the Euler-Lagrange equations for a elastic line on an oriented surface in the Galilean 3-dimensional space . Using the varia- tion method, we will try to give some characterization for the solution curve (the elastic lin…
New method weaves paper strips for designing curved surfaces with elasticity.
problem Designing general curved surfaces with geometrical elasticity.
method Shape optimization of paper strips using nonlinear elasticity theory.
result Demonstrated creation of catenoid and helicoid surfaces with 54 paper strips.
Unified treatment of elastic metrics for curves in any dimension.
problem Defining metrics on spaces of Euclidean curves for statistical analysis.
method Developing a unified approach to elastic metrics, extending results on existence of solutions and algorithms for computing distances and geodesics.
result Unified treatment of elastic metrics for all parameter choices, extending previous work.
We study the evolution of closed inextensible planar curves under a second order flow that decreases the -elastic energy. A short time existence result for is obtained via a minimizing movements method. For , that is in the case of the classic elastic energy, long-time existence is retrieve…
New elastic energy for irregular curves defined through polygonal approximations.
problem Defining elastic energy for irregular curves in any space dimension.
method Relaxation process with -rotation of inscribed polygonals, focusing on geometric curvature distribution.
result Energy finite if and only if curve's arc-length parameterization has second order summability.
On p-biharmonic curvesmath.DG
Study of -biharmonic curves and their properties.
problem Generalizing biharmonic curves to -biharmonic curves.
method Classification and analysis of -biharmonic curves on surfaces and space forms.
result Existence and stability of -biharmonic curves on closed surfaces.
Jellyfish existmath.DG
New jellyfish found in various flows.
problem Existence of geometrically distinct shapes in flows.
method Analyzing elastic, curve diffusion, and ideal flows.
result Infinitely many distinct shapes discovered.
The paper classifies and analyzes the stability of elastic curves with fixed endpoints.
problem Classification and stability of pinned elasticae.
method Critical points of the length-penalized elastic bending energy among planar curves with fixed endpoints.
result Explicit parametrization and classification of all critical points with a threshold parameter \(\hatλ \simeq 0.70107\).
A new algorithm computes elastic shape distances between curves efficiently.
problem Computing elastic shape distances between curves in high dimensions.
method Dynamic Programming for optimal diffeomorphisms and Kabsch-Umeyama algorithm for optimal rotation matrices.
result Efficient computation of elastic shape distances with improved efficiency for closed curves.
New method shortens and straightens curves, proving convergence and well-posedness.
problem Shortening and straightening of curves.
method Conceptual shift in curve shortening to tangent aligning, variational study of geometric flows.
result Proves convergence to a straight line and global well-posedness for various geometric flows.
We consider an obstacle problem for elastic curves with fixed ends. We attempt to extend the graph approach provided in [8]. More precisely, we investigate nonexistence of graph solutions for special obstacles and extend the class of admissible curves in a way that an existence result can be obtained by a penalization …
The chapter reviews metrics for comparing curves, focusing on quotient elastic and square root velocity metrics.
problem Comparing and analyzing shapes of curves.
method Construction and theoretical properties of quotient elastic metrics, special case of square root velocity metric, numerical approaches for estimation.
result Simplified expression for the square root velocity metric distance.
Gradient flow of elastic energy converges to elastica.
problem Optimizing closed curves to minimize elastic energy.
method Proving the existence of a unique global solution and convergence via Łojasiewicz--Simon gradient inequality.
result Convergence to elastica established for the -gradient flow of modified elastic energy.
New metrics for surface shapes incorporating curve properties.
problem Developing metrics for surface shape spaces.
method Incorporates geodesic and normal curvatures of curves on surfaces.
result Explicitly defined 6-parameter family of metrics.
Study finds non-CMC biconservative hypersurfaces in spheres, proving their existence but not embeddability.
problem Characterizing and proving the existence of non-CMC biconservative hypersurfaces in spheres.
method Analyzing -elastic curves of profile curves of biconservative rotational hypersurfaces in space forms.
result Existence of a discrete biparametric family of non-CMC closed biconservative hypersurfaces in , none of which can be embedded.
Sharp convergence rate for curvature stability in planar free elastic flow.
problem Stability of -circles under the planar free elastic flow.
method Improved closeness measurement via curvature scalar, leading to a sharp convergence rate.
result Sharp convergence rate for curvature stability in planar free elastic flow.