New property: polygons have a fixed dimension regardless of ambient space dimensions.
problem Understanding the dimension of polygon moduli spaces.
method Generalizing the square bending example to polygons of arbitrary edge lengths.
result There are only finitely many moduli spaces of polygons with given edge lengths, even as ambient dimension increases.
We study the moduli spaces of polygons in R^2 and R^3, identifying them with subquotients of 2-Grassmannians using a symplectic version of the Gel'fand-MacPherson correspondence. We show that the bending flows defined by Kapovich-Millson arise as a reduction of the Gel'fand-Cetlin system on the Grassmannian, and with t…
Softens tilings in 3D space, proving conjectures about polyhedral tilings.
problem Proving that all locally polyhedral tilings in 3D space can be softened.
method Developed a new edge-bending algorithm to prove the statement.
result Proved conjectures about polyhedral tilings in 3D space and the plane.
We evaluate the distribution learning capabilities of generative adversarial networks by testing them on synthetic datasets. The datasets include common distributions of points in Rn space and images containing polygons of various shapes and sizes. We find that by and large GANs fail to faithfully recreate point dat…
We study the symplectic geometry of the moduli spaces $M_r=M_r(\s^3)$ of closed n-gons with fixed side-lengths in the 3-sphere. We prove that these moduli spaces have symplectic structures obtained by reduction of the fusion product of n conjugacy classes in SU(2), denoted Crn, by the diagonal conjugation action …
We study the symplectic geometry of the moduli space of closed n-gons with fixed side-lengths in hyperbolic 3-space. We prove that these moduli spaces have a symplectic structure coming from Poisson Lie theory. We construct completely integrable systems on these moduli spaces by bending n-gons along their diagonals. Th…
Duality principle for approximation of geometrical objects (also known as Eudoxus exhaustion method) was extended and perfected by Archimedes in his famous tractate "Measurement of circle". The main idea of the approximation method by Archimedes is to construct a sequence of pairs of inscribed and circumscribed polygon…
Study on bending knots and energy changes in 3D space.
problem Understanding energy changes in knots under small deformations.
method Analyzes infinitesimal bending of knots and energy changes using Willmore and Mobius energies.
result Changes in energy under small deformations of knots have been quantified.
Proves the bending map is proper for hyperbolic 3-manifolds.
problem Properness of the bending map in hyperbolic 3-manifolds.
method Analyzes geometric properties and isotopy classes of homeomorphisms.
result Proving the bending map is proper for hyperbolic 3-manifolds.
Periodic surfaces have a limited number of bending modes, equal to their membrane modes.
problem Understanding the limitations of bending modes in periodic surfaces.
method Analyzing deformation modes of periodic, piecewise smooth, simply connected surfaces.
result Effective membrane modes and bending modes are orthogonal, limiting the total number of modes to 3.
Minimal surfaces can be transformed into others with unchanged bending content.
problem Understanding the deformation properties of minimal surfaces.
method Refined polar decomposition theorem to identify bending-neutral deformations.
result Every minimal surface can be transformed into another by a bending-neutral deformation.
The study explores isometric deformations of surfaces of translation.
problem Determine the ways surfaces of translation bend isometrically.
method Analyzes existence conditions and provides closed-form expressions for infinitesimal and finite bendings of surfaces of translation.
result Surfaces of translation admit various infinitesimal and finite bendings, including purely torsional and torsion-free.
Study bends 2D surfaces in 3D space using special equations.
problem Investigate infinitesimal bendings of 2D surfaces in 3D space.
method Use Bers-Vekua type equations and systems of differential equations with periodic coefficients.
result Construct bending fields for specific classes of 2D surfaces.
We examine the dependence of the deformation obtained by bending quasi-Fuchsian structures on the bending lamination. We show that when we consider bending quasi-Fuchsian structures on a closed surface, the conditions obtained by Epstein and Marden to relate weak convergence of arbitrary laminations to the convergence …
Study on bending deformations in hyperbolic manifolds, generalizing Johnson and Millson's work.
problem Understanding infinitesimal deformations in branched bending complexes.
method Defining branched bending deformations, giving lower bounds, and constructing examples.
result Lower bounds on the dimension of deformation spaces and examples of specific deformations.
Generalizes existence of bending laminations for Kleinian groups.
problem Existence of bending laminations for Kleinian surface groups.
method Generalization of Bonahon and Otal's proof to include geometrically infinite groups.
result Compactness of Kleinian groups realizing specific laminations.
The paper bends Riemannian manifolds to achieve metrics with negative Ricci curvature.
problem Achieving metrics with negative Ricci curvature on closed Riemannian manifolds.
method Solving a fully nonlinear equation to conformally bend the manifold.
result Metrics of quasi-negative Ricci curvature are conformal to metrics with negative Ricci curvature.
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.
We prove Thurston's bending measure conjecture for quasifuchsian once punctured torus groups. The conjecture states that the bending measures of the two components of the convex hull boundary uniquely determine the group.
Study finds how periodic surfaces can bend without stretching.
problem Understanding isometric deformations of periodic surfaces.
method Characterization of isometric deformations using a constraint derived from Gauss theorem.
result Relates surface stretching to bending and twisting.
Bounds projective structure norms by bending lamination lengths.
problem Bounding the L2-norm of projective structures. method Using the Thurston parameterization and Krasnov-Schlenker's W-volume theory. result Upper bounds on L2-norm of holomorphic quadratic differential by the length of bending lamination. This paper proves unique determination of certain non-quasi-Fuchsian manifolds by their end structure and bending lamination.
problem Unique determination of non-quasi-Fuchsian manifolds by their end structure and bending lamination.
method Analysis of the end structure (parabolic locus, ending laminations, conformal structures) and bending lamination.
result Non-quasi-Fuchsian manifolds are uniquely determined by their end structure and bending lamination.
Study surface subgroups acting on projective space, finding bending laminations and spheres.
problem Surface subgroups acting on RP3 with coaffine representations. method Stratification of convex core boundary, bending laminations, and analysis of holonomy.
result Projectivization of bending data space is a sphere of dimension 6g−7. Proposes a new network to improve nuclei segmentation in histopathology images.
problem Challenges in separating overlapped nuclei in histopathology images.
method Introduces a bending loss regularized network to minimize contour points with large curvatures.
result Outperforms six state-of-the-art approaches on five quantitative metrics.
The bending map of a hyperbolic 3-manifold maps a convex cocompact hyperbolic metric on a hyperbolic 3-manifold with boundary to its bending measured geodesic lamination. In the present paper we study the extension of this map to the space of geometrically finite hyperbolic metrics. We introduce a relationship on the s…
New bounds link Schwarzian derivative to hyperbolic geometry.
problem Quantify the relationship between Schwarzian derivative and hyperbolic geometry.
method Established explicit quantitative bounds between Schwarzian norm and bending norm.
result Explicit bounds on bending norm for univalent maps with small Schwarzian norm.
We investigate the elastic behavior of knotted loops of springy wire. To this end we minimize the classic bending energy Ebend=∫κ2 together with a small multiple of ropelength R=length/thickness in order to penalize selfintersection. Our main objective is to characterize elastic…
New solutions found for bending of flat surfaces and origami structures.
problem Understanding the energy-efficient bending modes of origami tessellations and corrugated shells.
method Direct construction of closed-form solutions for surfaces of translation.
result Three inextensional modes identified for surfaces of translation, including stretching, bending, and twisting.
Twisted SL2C local systems on surfaces of finite type appear often in geometry and physics. Most of them arise geometrically as local systems of charts for pleated hyperbolic structures. Bonahon and Thurston's "shear-bend coordinates" parameterize these local systems of charts. On a surface …
Researchers find a surface with minimum bending energy for any genus and isoperimetric ratio.
problem Finding surfaces with minimum bending energy for given genus and isoperimetric ratio.
method Gluing catenoidal bridges to a singular solution of the Willmore equation on a punctured sphere.
result Existence of a surface with minimum bending energy for any genus and isoperimetric ratio.
We study deformations of complex hyperbolic surfaces which furnish the simplest examples of: (i) negatively curved Kähler manifolds and (ii) negatively curved Riemannian manifolds not having {\it constant} curvature. Although such complex surfaces may share the rigidity of quaternionic/octionic hyperbolic manifolds, ou…
A mapping bends Teichmüller spaces into character varieties, preserving symplectic structure.
problem Mapping Fricke-Teichmüller space to character variety of surface representations.
method Bending Fuchsian representations along a fixed measured lamination, proving equivariant symplectic embedding and properness.
result Continuous extension of bending map to Thurston boundary and geometric complexification.
New methods classify convex lattice polygons for affine dimers.
problem Not all convex lattice polygons are characteristic polygons of affine dimers.
method General constructions and algorithm for finding affine dimers with prescribed polygons.
result All lattice triangles, generalised parallelograms, and polygons of genus at most two admit an affine dimer.
Let M and N be compact smooth oriented Riemannian n-manifolds without boundary embedded in Rn+1. Several problems about minimal distortion bending and morphing of M to N are posed. Cost functionals that measure distortion due to stretching or bending produced by a diffeomorphism h:M→N are …
The Hopf fibration has inspired any number of geometric structures in physical systems, in particular in chiral liquid crystalline materials. Because the Hopf fibration lives on the three sphere, S3, some method of projection or distortion must be employed to realize textures in flat space. Here, we explore…
The pentagram map's limit point is related to infinitesimal perturbations of polygons.
problem Understanding the limit point of the pentagram map and its relation to polygon perturbations.
method Interpreting Glick's operator as the infinitesimal monodromy of a polygon.
result Glick's operator measures the extent to which a perturbed polygon does not close up.
Critical trajectories in a sphere are found for a specific bending functional.
problem Finding closed trajectories in a sphere for a specific bending functional.
method Existence of infinitely many closed trajectories shown for a given Lagrange multiplier.
result Existence of closed trajectories dependent on a pair of relatively prime natural numbers.
General area-preserving motion of polygonal curves is formulated as a system of ODEs. Solution polygonal curves belong to a prescribed polygonal class, which is similar to the admissible class used in the crystalline curvature flow. The ODEs are discretized implicitly in time keeping a given constant area speed while s…
Study finds finitely many non-congruent polygonal domains with same Steklov spectrum.
problem Inverse Steklov problem on convex polygons.
method Analysis of Steklov eigenvalues and isoperimetric bounds.
result For almost all convex polygonal domains, there exist at most finitely many non-congruent domains with the same Steklov spectrum.
Constructs minimal surfaces by gluing saddle towers with Scherk ends.
problem Embedding minimal surfaces with Scherk ends.
method Gluing saddle towers with prescribed phase differences, analyzing slight bendings.
result Correctly identifies scenarios where constructed surfaces are embedded.
The paper classifies vertices in planar polygons formed by convex domains.
problem Classifying vertices in planar polygons formed by convex domains.
method Analyzing polygons formed by homothets and translates of a convex domain.
result The number of singular boundary points in a C-polygon is between n and 2(n−1)+m for a strictly convex domain with m singular boundary points. In this paper, we discuss centroaffine geometry of polygons in 3-space. For a polygon X that is locally convex with respect to an origin together with a transversal vector field U, we define the centroaffine dual pair (Y,V) similarly to [6]. We prove that vertices of (X,U) correspond to flattening points for …
We establish a fundamental connection between smooth and polygonal knot energies, showing that the Minimum Distance Energy for polygons inscribed in a smooth knot converges to the Moebius Energy of the smooth knot as the polygons converge to the smooth knot. However, the polygons must converge in a ``nice'' way, and th…
The pentagram map takes a planar polygon P to a polygon P′ whose vertices are the intersection points of consecutive shortest diagonals of P. This map is known to interact nicely with Poncelet polygons, i.e. polygons which are simultaneously inscribed in a conic and circumscribed about a conic. A theorem of R. Sc…
Simple rectilinear polygons (i.e. rectilinear polygons without holes or cutpoints) can be regarded as finite rectangular cell complexes coordinatized by two finite dendrons. The intrinsic l1-metric is thus inherited from the product of the two finite dendrons via an isometric embedding. The rectangular cell complexe…
Optimal Reeb graphs identified for polygon decomposition.
problem Investigating the topological structure of planar polygon decomposition.
method Using oriented Reeb graphs with a marked vertex for height functions.
result Described all possible optimal Reeb graphs for specific polygon configurations.
Study on Poncelet polygons' centers and circumcenters in various geometries.
problem Understanding Poncelet polygons' geometric centers in different geometries.
method Analyzing the Circumcenter of Mass and Center of Mass of Poncelet polygons, proving Dan Reznik's invariants, and exploring spherical geometry.
result Proof of Dan Reznik's invariants for billiard trajectories and insights into Poncelet polygons' centers in spherical geometry.
The study proves analogues of the discrete isoperimetric inequality in hyperbolic geometry.
problem Finding the minimum perimeter for polygons with a fixed area in hyperbolic geometry.
method Proving analogues of the discrete isoperimetric inequality for cyclic and tangential polygons in hyperbolic geometry, considering both single and multiple polygons.
result Established two versions of the isoperimetric inequality for multiple polygons in hyperbolic geometry with certain area or perimeter restrictions.