Paper constructs moving frame for centroaffine curves to identify and analyze polygon flows.
problem Discriminate and analyze stability of polygon flows.
method Constructs moving frame and invariants for discrete centroaffine curves using centroaffine curvatures and torsions.
result Identifies stable and periodically stable discrete curves using centroaffine curvatures and torsions.
The paper studies self-Bäcklund curves in centroaffine geometry using elliptic functions.
problem Understanding self-Bäcklund curves in centroaffine geometry.
method Description of general properties and detailed analysis using elliptic functions.
result Provides a detailed description of self-Bäcklund centroaffine curves in terms of elliptic functions.
The paper studies discrete centroaffine surfaces in 3D space.
problem Understanding centroaffine invariants and convexity of discrete surfaces.
method Developed structure equations and integrable systems for discrete centroaffine surfaces. Calculated centroaffine invariants and analyzed Laplacian operator.
result Obtained centroaffine invariants and studied their implications on surface convexity.
The paper quantifies fractal curves using centroaffine curvatures.
problem Quantifying the irregularities of fractal curves.
method Using moving frame and centroaffine curvatures.
result Fractal curves can be described by a sequence of affine curvatures.
We construct integrable hierarchies of flows for curves in centroaffine R3 through a natural pre-symplectic structure on the space of closed unparametrized starlike curves. We show that the induced evolution equations for the differential invariants are closely connected with the Boussinesq hierarchy, and …
Complete classification of centroaffine hypersurfaces with parallel cubic form.
problem Characterizing centroaffine hypersurfaces with specific geometric properties.
method Analyzing hypersurfaces with respect to the Levi-Civita connection of the centroaffine metric.
result A complete classification of locally strongly convex centroaffine hypersurfaces with parallel cubic form.
New maximal surfaces solve Bernstein problems.
problem Bernstein problems in centroaffine geometry.
method Calabi affine maximal surfaces and orthonormal frame fields.
result Complete centroaffine extremal hypersurfaces solve all Bernstein problems.
We use Cartan's method of moving frames to compute a complete set of local invariants for nondegenerate, 2-dimensional centroaffine surfaces in R5∖{0} with nondegenerate centroaffine metric. We then give a complete classification of all homogeneous centroaffine surfaces in this class.
The paper proves ellipsoids are the only centroaffine Tchebychev hyperovaloids.
problem Generalizing Blaschke and Deicke's theorem to centroaffine differential geometry.
method Characterized hypersurfaces by a closed conformal vector field and used properties of Riemannian manifolds.
result Ellipsoids are the only centroaffine Tchebychev hyperovaloids.
The paper explores centroaffine geometry of polygons and their duals.
problem Understanding centroaffine dual pairs of spatial polygons.
method Defining centroaffine dual pairs and proving properties of polygon duals.
result Constant curvature polygons are dual to planar polygons.
Paper establishes an optimal inequality for convex hypersurfaces.
problem Optimal inequality for locally strongly convex centroaffine hypersurfaces.
method Used covariant derivatives of difference tensor and Tchebychev vector field.
result Complete classification of hypersurfaces realizing equality in inequality.
This paper is concerned with the completeness (with respect to the centroaffine metric) of hyperbolic centroaffine hypersurfaces which are closed in the ambient vector space. We show that completeness holds under generic regularity conditions on the boundary of the convex cone generated by the hypersurface. The main re…
It is demonstrated that hypersurfaces with a flat centroaffine metric are governed by a system of nonlinear PDEs known as the equations of associativity of 2-dimensional topological field theory.
This paper classifies hypersurfaces in n+1 with parallel Fubini-Pick form.
problem Classifying hypersurfaces with parallel Fubini-Pick form in \(\mathbb{R}^{n+1}\).
method Defining a generalized Calabi product and proving decomposition theorems.
result Complete classification of Calabi hypersurfaces in \(\mathbb{R}^{n+1}\) with parallel Fubini-Pick form.
In this paper, we study strictly convex affine hypersurfaces centroaffinely congruent to their centre map, in the case when the shape operator has two distinct eigenvalues: one of multiplicity 1, and one nonzero of multiplicity n-1. We show how to construct them from (n-1)-dimensional affine hyperspheres.
Discrete model of curve deformation using discrete nonlinear Schrödinger equation.
problem Deformation of discrete space curves.
method Discrete analogue of the local induction equation using the discrete nonlinear Schrödinger equation.
result Explicit formulas for smooth and discrete curves in terms of τ functions of the two-component KP hierarchy.
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.
We construct explicit solutions to continuous motion of discrete plane curves described by a semi-discrete potential modified KdV equation. Explicit formulas in terms the τ function are presented. Bäcklund transformations of the discrete curves are also discussed. We finally consider the continuous limit of discrete …
The paper introduces log-aesthetic curves and their integrable discretization.
problem Characterizing and discretizing log-aesthetic curves.
method Similarity geometry, integrable Burgers equation, variational principles.
result Proposed variational principle and discretization preserving integrable structure.
New method solves discrete mKdV equation from curve motions.
problem Solving the discrete potential mKdV equation.
method Discrete Darboux transformation.
result Efficient route to discrete mKdV equation.
Discretization of curves is an ancient topic. Even discretization of curves with an eye toward differential geometry is over a century old. However there is no general theory or methodology in the literature, despite the ubiquitous use of discrete curves in mathematics and science. There are conflicting definitions of …
New geometric transformations link discrete and continuous curve motions.
problem Establishing a connection between discrete and continuous curve motions.
method Infinitesimal Darboux transformations of smooth curves.
result Alternate geometric interpretation for semi-discrete mKdV equation.
In this paper, we consider the discrete deformation of the discrete space curves with constant torsion described by the discrete mKdV or the discrete sine-Gordon equations, and show that it is formulated as the torsion-preserving equidistant deformation on the osculating plane which satisfies the isoperimetric conditio…
The paper studies stability of discrete planar curves using variational methods.
problem Stability of discrete planar curves under area constraints.
method Unified interpretation of discrete curvatures, determination of equilibrium curves, stability analysis.
result Equilibrium curves for the length functional under area-constraint conditions are determined and their stability is studied.
Every discrete subset in a complex domain is in a complex curve.
problem Embedding discrete subsets in complex domains.
method Proving every closed discrete subset is in a complex curve with any topology.
result Closed discrete subsets are contained in complex curves with any topology.
The paper proves Γ-convergence of discrete tangent-point energies to continuous energies and ropelength, with applications to biarc curves.
problem Proving convergence of discrete tangent-point energies to continuous energies and ropelength.
method Using biarc curves and interpolation, the paper proves Γ-convergence of discretized tangent-point energies to the continuous tangent-point energies and ropelength functional. result Discrete almost minimizing biarc curves converge to ropelength minimizers and minimizers of continuous tangent-point energies.
New method linearizes Darboux transformations of discrete curves.
problem Linearizing Darboux transformations of discrete curves.
method Expressing Darboux transformations as parallel sections of discrete connections in quaternionic formalism.
result Closed-form discrete parametrisations of all Darboux transforms and bicycle correspondences.
We construct explicit solutions to the discrete motion of discrete plane curves that has been introduced by one of the authors recently. Explicit formulas in terms the τ function are presented. Transformation theory of the motions of both smooth and discrete curves is developed simultaneously.
Bäcklund transformations for smooth and ``space discrete'' Hashimoto surfaces are discussed and a geometric interpretation is given. It is shown that the complex curvature of a discrete space curve evolves with the discrete nonlinear Schrödinger equation (NLSE) of Ablowitz and Ladik, when the curve evolves with the Has…
Study curvature and torsion from cross-ratios in discrete curves.
problem Define curvature and torsion for discrete curves using cross-ratios.
method Use Möbius invariant point-insertion-rule to construct circles and express torsion using cross-ratio.
result Discrete curvature and torsion defined using cross-ratios converge to smooth curvature and torsion as sampling density increases.
In this paper we investigate flows on discrete curves in $\C^2$, $\CP^1$, and $\C$. A novel interpretation of the one dimensional Toda lattice hierarchy and reductions thereof as flows on discrete curves will be given.
The study proves a discrete version of Segre's theorem for polygonal curves.
problem Proving a discrete analog of a four-vertex theorem for spherical curves.
method Using the concept of discrete tangent indicatrix of a polygon.
result A polygon with at least four vertices and a non-self-intersecting discrete tangent indicatrix has at least four flattenings.
Study on Brownian motion on discrete curve spaces, proving stochastic completeness.
problem Analyzing Brownian motion on spaces of discrete curves.
method Introduced and studied Brownian motion on spaces of discrete regular curves with Sobolev-type metrics.
result All geodesically complete spaces of discrete regular curves are stochastically complete.
New geometric mechanics approach to elastic curves.
problem Understanding elastic curves in mechanics and geometry.
method Developed a new geometric mechanics perspective on elastic curves.
result Elastic curves are critical points of length under fixed area and volume constraints.
Selberg's Lemma fails for certain curved manifolds.
problem Applying Selberg's Lemma to negatively curved Hadamard manifolds.
method Proving the failure of Selberg's Lemma for specific groups of manifolds.
result Selberg's Lemma does not hold for discrete isometry groups of negatively curved Hadamard manifolds.
The paper analyzes discrete approximations to minimize curve length in Euclidean space.
problem Minimizing the length of curves between two sets in Euclidean space.
method Finite differences and numerical integration for discrete approximations.
result The squared length of the reconstructed curve converges to the squared minimal length with rate O(N−1/2). Study of discrete Koenigs nets and their properties.
problem Characterization and properties of discrete Koenigs nets.
method Generalization of inscribed conics to inscribed quadrics and study of Koenigs d-grids.
result Established a bijection between Koenigs d-grids and pairs of discrete autoconjugate curves.
The study of equal-volume polygons in 3D space and their affine invariants.
problem Estimating projective invariants of planar curves.
method Developing a theory of discrete affine invariants from equal-volume polygons.
result Equal-volume polygons can be used to estimate projective invariants of a planar curve.
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 p-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.
New metrics on curve spaces improve shape analysis.
problem Discretization of curve spaces and metric completeness.
method Sobolev metrics on discrete regular curves, completeness analysis.
result The finite-dimensional Riemannian manifolds are complete.
Flows on (or variations of) discrete curves in R2 give rise to flows on a subalgebra of functions on that curve. For a special choice of flows and a certain subalgebra this is described by the Toda lattice hierachy. In the paper it is shown that the canonical symplectic structure on R2N, which can be interpre…
The discrete Nahm equations, a system of matrix valued difference equations, arose in the work of Braam and Austin on half-integral mass hyperbolic monopoles. We show that the discrete Nahm equations are completely integrable in a natural sense: to any solution we can associate a spectral curve and a holomorphic line-b…
Unified approach to discrete and smooth isoperimetric inequalities of arbitrary order.
problem Finding higher order isoperimetric inequalities for both discrete and smooth curves.
method Unified approach via Fourier analysis of linear operators.
result Unified upper and lower bounds for isoperimetric deficit in smooth curves.
We develop the structure theory of full isometry groups of locally compact non-positively curved metric spaces. Amongst the discussed themes are de Rham decompositions, normal subgroup structure and characterising properties of symmetric spaces and Bruhat--Tits buildings. Applications to discrete groups and further dev…
A Darboux transformation for polarized space curves is introduced and its properties are studied, in particular, Bianchi permutability. Semi-discrete isothermic surfaces are described as sequences of Darboux transforms of polarized curves in the conformal n-sphere and their transformation theory is studied. Semi-discre…
Paper finds optimal matching between curves on manifolds using geodesics.
problem Comparing shapes of manifold-valued curves.
method Introduced a simple algorithm using canonical decomposition and geodesics.
result Optimal matching between curves found using geodesic computation.
The Dirichlet Laplacian in curved tubes of arbitrary cross-section rotating with respect to the Tang frame along infinite curves in Euclidean spaces of arbitrary dimension is investigated. If the reference curve is not straight and its curvatures vanish at infinity, we prove that the essential spectrum as a set coincid…
Paper studies how discrete space curves with constant torsion deform to model linkage motions.
problem Modeling and understanding the motion of discrete space curves with constant torsion.
method Using semi-discrete mKdV equations to describe the motion of discrete space curves.
result The motion of discrete space curves is governed by semi-discrete mKdV equations.