Study on CR curves in 3-sphere, focusing on critical curves integration and existence.
problem Addressing the integration and existence of critical curves in the CR 3-sphere.
method Provided a procedure for the explicit integration of general critical curves and characterized closed curves.
result Existence of infinite countably many closed critical curves.
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.
Kirchhoff energy is a classical functional on the space of arclength-parameterized framed curves whose critical points approximate configurations of springy elastic rods. We introduce a generalized functional on the space of framed curves of arbitrary parameterization, which model rods with axial stretch or cross-secti…
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 \(\hatλ \simeq 0.70107\).
Study variational properties of curves in half-plane with area constraints.
problem Characterize critical points of inverse mean curvature.
method Variational analysis of curves with boundary constraints.
result Existence and stability of critical points with prescribed area.
Study on curve diffusion flows with scale-critical curvature term.
problem Analyzing stability of curve diffusion flows with scale-critical curvature.
method Introduced and studied a one-parameter family of curve diffusion flows with a scale-critical cubic curvature term. Analyzed dynamical stability of homothetic circles using variational methods.
result Established that any small perturbation of an ω-fold circle monotonically approaches the unit ω-circle after rescaling, translation, and reparametrisation. Classifies soap film surfaces with vertical potentials.
problem Classifying soap film surfaces with vertical potentials.
method Variational characterization of n-elastic curves. result Obtains a full description of n-elastic curves. Study on cr-invariant variational problem for Legendrian curves in 3-sphere.
problem Lower-order cr-invariant variational problem for Legendrian curves in 3-sphere.
method Deduced Euler-Lagrange equations, investigated closed critical curves, characterized non-constant cr-curvature curves, proved cr-equivalence classes correspondence to rational points.
result Closed critical curves with non-constant cr-curvature are characterized and their cr-equivalence classes are in one-to-one correspondence with rational points of a connected planar domain.
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.
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 L2-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 of critical tori for mean curvature energies in Killing submersions.
problem Analyzing surface energies in Killing submersions.
method Symmetry reduction and binormal evolution of critical curves.
result Construction of vertical tori critical for mean curvature energies.
Critical graphs of quadratic differentials equidistribute in moduli space.
problem Distribution of critical graphs in moduli space.
method Study of Jenkins-Strebel differentials and their critical graphs.
result Critical graphs equidistribute to the Kontsevich measure.
In this paper, we apply classical energy principles to Euler elasticae, i.e., closed C^2 curves in the plane supplied with the Euler functional U (the integral of the square of the curvature along the curve). We study the critical points of U, find the shapes of the curves corresponding to these critical points and sho…
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.
A theorem connects integral of second-order derivatives to function rise.
problem Understanding the integral of second-order derivatives over regions.
method Proves integral proportional to function rise over specified regions.
result Integral of second-order derivatives equals rise in function value.
New proof for certain groups in higher dimensions.
problem Properties of discrete subgroups in higher dimensions.
method Proving convex-cocompactness for specific groups.
result Finitely generated Kleinian groups with small critical exponent are convex-cocompact.
Investigates energy minimizers and critical points of scale-invariant tangent-point energies for knots.
problem Finding and characterizing minimizers and critical points of scale-invariant tangent-point energies for closed curves.
method Develops convergence and regularity theories based on fractional Sobolev spaces and new energy functionals.
result Minimizing sequences converge to locally critical embeddings in all but finitely many points, and locally critical embeddings are regular.
The normal map of curves is analyzed as a vector field on a cylinder.
problem Understanding the geometric properties of normal maps and their vector field interpretation.
method Interpreting critical points geometrically, studying Poincaré index, projecting to sphere, and analyzing winding and rotation indices.
result Counting theorems regarding winding and rotation indices of curves and their evolutes are proven.
We consider variation of energy of the light-like particle in Riemann space-time, find lagrangian, canonical momenta and forces. Equations of the critical curve are obtained by the nonzero energy integral variation in accordance with principles of the calculus of variations in mechanics. This method is shown to not lea…
Develops finite-gap solutions for Pohlmeyer--Lund--Regge equation and Lund--Regge curve evolution.
problem Solving the Pohlmeyer--Lund--Regge equation and understanding Lund--Regge curve evolution.
method Finite-gap construction using hyperelliptic spectral data, Baker--Akhiezer function, and SU(2)-frame. result Explicit theta-quotient formula for PLR solutions and criteria for Lund--Regge curve evolution.
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.
Given two Jordan curves in a Riemannian manifold, a minimal surface of annulus type bounded by these curves is described as the harmonic extension of a critical point of some functional (the Dirichlet integral) in a certain space of boundary parametrizations. The H2,2-regularity of the minimal surface of annulus t…
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 paper classifies flows of ancient curves in 2D space.
problem Classifying closed convex flows by curvature powers.
method Sub-affine-critical powers of curvature for flow classification.
result Ancient flows converge exponentially to smooth shrinkers.
The paper classifies surfaces with constant skew curvature in 3-space forms.
problem Classifying surfaces with constant skew curvature in 3-space forms.
method Variational characterization and flow of binormal vector field.
result Classification of rotational surfaces with constant skew curvature.
The paper extends Euler's problem to hyperbolic and spherical planes.
problem Extending Euler's problem to hyperbolic and spherical planes.
method Characterizing critical points of moment of inertia energy in hyperbolic and spherical planes.
result Closed stationary curves in hyperbolic plane are circles centered at N.
Characterizes solutions to Z-critical equations on surfaces using effective conditions.
problem Characterizing solutions to Z-critical equations on compact Kähler surfaces.
method Uses effective conditions and Picard number bounds to characterize solutions.
result Characterizes optimally destabilizing curves for Donaldson's J-equation and deformed Hermitian Yang-Mills equation.
After having given the general variational formula for the functionals indicated in the title, the critical points of the integral of the equi-affine curvature under area constraint and the critical points of the full-affine arc-length are studied in greater detail.
Paper finds relations between Willmore-type energies, weighted areas, and vertical potential energies for cylindrical critical points.
problem Tackles relations between three types of energy functions for cylindrical critical points.
method Uses differential equations and critical point analysis for Willmore-type energies and weighted areas.
result Generating curves coincide for Willmore-type energies and weighted areas, and similar results hold for Willmore-type energies and vertical potential energies.
For any geodesic current we associated a quasi-metric space. For a subclass of geodesic currents, called filling, it defines a metric and we study the critical exponent associated to this space. We show that is is equal to the exponential growth rate of the intersection function for closed curves.
Given a Riemannian metric on the 2-sphere, sweep the 2-sphere out by a continuous one-parameter family of closed curves starting and ending at point curves. Pull the sweepout tight by, in a continuous way, pulling each curve as tight as possible yet preserving the sweepout. We show the following useful property (see Th…
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
The ropelength problem asks for the minimum-length configuration of a knotted diameter-one tube embedded in Euclidean three-space. The core curve of such a tube is called a tight knot, and its length is a knot invariant measuring complexity. In terms of the core curve, the thickness constraint has two parts: an upper b…
This paper concerns with the asymptotic behavior of complete non-compact convex curves embedded in R2 under the α-curve shortening flow for exponents α>21. We show that any such curve having in addition its two ends asymptotic to two parallel lines, converges under α-curve shortening flow to the …
Study on curve shortening flow with boundary conditions, proving convergence or contraction.
problem Analyzing curve shortening flow with free boundaries.
method Introduced a reflected chord-arc profile and obtained chord-arc estimates.
result Proved that flows either converge to a critical chord or contract to a round half-point.
We prove the shifting theorems of the critical groups of critical points and critical orbits for the energy functionals of Finsler metrics on Hilbert manifolds of H1-curves, and two splitting lemmas for the functionals on Banach manifolds of C1-curves. Two results on critical groups of iterated closed geodesics a…
We uncover some connections between the topology of a complete Riemannian surface M and the minimum number of vertices, i.e., critical points of geodesic curvature, of closed curves in M. In particular we show that the space forms with finite fundamental group are the only surfaces in which every simple closed curve ha…
Flow of curves with curvature and forcing vector field exists.
problem Existence of a curve flow with curvature and forcing.
method Proved existence through Brakke motion law.
result Non-trivial flow of curves exists through singularities.
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…
Study on Gaussian random fields' singularities on manifolds.
problem Understanding singularities of Gaussian random fields on manifolds.
method Computed expected values of singularities under various conditions.
result Explicit formulae for singularities under different constraints.
We establish the splitting lemmas (or generalized Morse lemmas) for the energy functionals of Finsler metrics on the natural Hilbert manifolds of H1-curves around a critical point or a critical R1 orbit of a Finsler isometry invariant closed geodesic. They are the desired generalization on Finsler manifolds of t…
Characterizes curves for minimal surfaces in de Sitter space.
problem Minimal surfaces in de Sitter space.
method Variational problem to find critical points of center of mass.
result Curves are critical points of center of mass.
The paper proves properties of curves in Riemannian manifolds.
problem Characterizing curves in Riemannian manifolds.
method Analyzing locally minimizing and weak geodesics.
result Locally minimizing curves are weak geodesics under certain conditions.
Extends Euler's problem to Lorentz-Minkowski plane.
problem Finding critical points of moment of inertia in Lorentz-Minkowski space.
method Explicit solutions for stationary curves, symmetries, inversions, and energy maximization.
result Explicit solutions for stationary spacelike and timelike curves, and methods to transform between them.
The paper studies the convergence of elastic flows of curves into manifolds, proving smooth convergence under certain conditions.
problem The convergence of elastic flows of curves into manifolds.
method Parabolic estimates and Lojasiewicz-Simon gradient inequality.
result Smooth convergence of the flow to critical points under specific conditions.
We propose a generalization of tropical curves by dropping the rationality and integrality requirements while preserving the balancing condition. An interpretation of such curves as critical points of a certain quadratic functional allows us to settle the existence and uniqueness problem. The machinery of dual polygons…
New insights into stability of special curves on spheres.
problem Stability of closed p-elastic curves on spheres. method Analytical proof and construction of curves.
result All closed spherical p-elastic curves for p∈(0,1) are unstable. Study spherical curves with curvature dependent on distance to a great circle.
problem Understanding spherical curves with curvature dependent on distance to a great circle.
method Introducing spherical angular momentum, characterizing known curves, finding new families, and obtaining arc length parametrizations.
result New families of spherical curves with intrinsic equations in elementary or Jacobi elliptic functions.