Constructs infinitely many length equivalent curves using Goldman bracket.
problem Finding infinitely many length equivalent curves in a surface.
method Using the Goldman bracket between curves and their intersections.
result Constructs infinitely many pairs of length equivalent curves.
Two free homotopy classes of closed curves in an orientable surface with negative Euler characteristic are said to be length equivalent if for any hyperbolic structure on the surface, the length of the geodesic in one class is equal to the length of the geodesic in the other class. We show that there are elements in th…
Study minima of geodesic lengths for specific curves on surfaces.
problem Finding the shortest geodesic paths on surfaces.
method Using curves related to dessins d'enfants and Grothendieck-Belyi surfaces.
result Minima of geodesic lengths are achieved on Riemann surfaces defined over number fields.
The Euclidean cone metrics coming from q-differentials on a closed surface of genus g > 1 define an equivalence relation on homotopy classes of closed curves declaring two to be equivalent if they have the equal length in every such metric. We prove an analog of the result of Randol for hyperbolic metrics (building on …
The paper connects curves on a light cone to KdV equations.
problem Understanding differential invariants of curves on a light cone.
method Poisson equivalence and centro-affine action of Lorentzian group.
result Solutions of KdV equations as flows of curves on the cone.
Study conditions for curvature functions of closed planar curves.
problem Conditions for curvature functions of closed planar curves.
method Equivalent conditions and periodic behaviors shown; explicit construction of pairs.
result Characterization of curvature functions and limitations of 4-vertex theorem.
In this paper, the parallel transport frames over non-lightlike curves in Minkowski 3-space are introduced. Evolution equations of these frames with respect to arc length and time are calculated over the space of these curves. Then the equivalence of the non-linear Schrödinger equation and non-linear heat system to the…
The paper extends deformation theory for curves of fixed degree in graded manifolds.
problem Computing the first variation of length functionals for curves of fixed degree.
method Analyzes curves in graded manifolds with Riemannian metrics and uses differential equations.
result Provides a sufficient condition for deforming curves of fixed degree.
Study on Lorentzian spaces with curvature bounds, proving comparison theorems.
problem Understanding curvature bounds in Lorentzian spaces.
method Introduced normalized angle for Lorentzian pre-length spaces, proving comparison theorems.
result Established local Lorentzian Toponogov theorem and Alexandrov convexity property.
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.
The flow of curves in Minkowski plane converges to a specific shape.
problem Analyzing curve diffusion in Minkowski plane.
method Anisotropic polyharmonic curve flow.
result Closed curves converge to a homothetic rescaling of the isoperimetrix.
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.
New invariant links graph structure to tropical curve properties.
problem Understanding graph and curve minor structures.
method Defined Ceresa-Zharkov class for graphs, related to tropical curves.
result Ceresa-Zharkov class is zero for hyperelliptic graphs.
Undergrad project: Shows geodesics coincide in Heisenberg group under two metrics.
problem Identifying geodesics in Heisenberg group under two metrics.
method Examined Heisenberg group H1 with Koranyi- and Carnot-Caratheodory metrics.
result Geodesics coincide for both metrics in Heisenberg group.
New spectra defined for metric spaces, extending existing covering spectrum.
problem Characterizing and comparing spectra for metric spaces.
method Defining and measuring 'entourage covers' to derive new spectra.
result New spectra (ECS, ES) extend existing covering spectrum (CS) and have useful properties.
Two Riemannian manifolds are called eigenvalue equivalent when their sets of eigenvalues of the Laplace-Beltrami operator are equal (ignoring multiplicities). They are (primitive) length equivalent when the sets of lengths of their (primitive) closed geodesics are equal. We give a general construction of eigenvalue equ…
We investigate the cross ratio for closed negatively curved manifolds. As one of several applications, we obtain that for two such homotopy equivalent manifolds M and N, the following is true : If M and N have the same marked length spectrum and if the Anosov splitting for M is C^1 then M and N have the same volume.
New curves generalize flat metrics from quadratic to q-differentials.
problem Determining flat metrics from curve lengths.
method Introduced q-simple curves to generalize results from quadratic to q-differentials.
result Lengths of q-simple curves uniquely determine non-positively curved Euclidean cone metrics induced by q-differentials.
Curves converge to circles under length constraints.
problem Understanding curve convergence under length constraints.
method Length-constrained curve diffusion to analyze curve behavior over time.
result Curves converge to circles in infinite time with exponential convergence.
Example of divergent horocycle in Teichmüller space.
problem Characterizing convergence in Teichmüller spaces.
method Constructing a specific curve and horocycle in a punctured sphere, then generalizing.
result Found a divergent horocycle in Teichmüller spaces of complex dimension greater than one.
The paper studies a curve flow preserving anisotropic length for convex curves, leading to a homothetic limit.
problem Anisotropic length preservation in curve deformation.
method A curve flow that maintains anisotropic length, analyzed for convex closed curves.
result Convex curves evolve to homothetic limits of Wulff shapes as time approaches infinity.
Timelike curves in Lorentzian length spaces have a total curvature notion that agrees with smooth curves.
problem Estimating the length of timelike curves in Lorentzian length spaces.
method Introducing a synthetic timelike total curvature notion.
result Proving timelike curves of finite total curvature are rectifiable.
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 L2-gradient flow for Euler's elastic energy. result An appropriate rescaling of initial curves geometrically close to circles converges to a unique round circle.
Proves contractibility of a curve traced multiple times under certain length constraints.
problem Contractibility of a curve traced multiple times under length constraints.
method Analyzes simple closed curves on Riemannian manifolds and their homotopies.
result If a multiple-traced curve is contractible, the original curve is contractible under similar length constraints.
The paper studies area-preserving and length-preserving inverse curvature flow for planar curves with singularities.
problem Investigating the evolution of planar curves with singularities under area-preserving and length-preserving inverse curvature flow.
method Area-preserving and length-preserving inverse curvature flow for ℓ-convex Legendre curves. result The flow results in a circle for ℓ-convex Legendre curves, providing geometric inequalities. Paper introduces length measures for curves and convex shapes, proving isoperimetric and distance properties.
problem Characterizing and comparing convex shapes using length measures.
method Developed length measures for curves and convex shapes, derived properties, and introduced a new distance metric.
result Unique convex curve maximizes signed area among curves with same length measure.
New method simplifies ideal curve flow with length constraint.
problem Analyzing ideal curve flow with length constraint.
method Introduced length constraint to simplify sixth order curvature flow.
result Flow exists for all time and converges to a round circle.
Study on stable translation lengths of surface homeomorphisms and their approximations.
problem Understanding stable translation lengths of homeomorphisms and their finite approximations.
method Comparing stable translation lengths of homeomorphisms and their finite approximations on curve graphs.
result Stable translation length of homeomorphisms with dense periodic points equals the supremum of their approximations.
Anosov maps on torus curve graphs have positive integer translation lengths.
problem Understanding the translation lengths of Anosov maps on curve graphs of tori.
method Constructive proof and algorithm for calculating exact translation lengths.
result The stable translation length of an Anosov map on the curve graph is always a positive integer.
Flow on curves in inversive geometry converges to loxodromics.
problem Gradient flow for curve length in inversive geometry.
method Invariant gradient flow for invariant length functional.
result Solutions exist for all time and converge to loxodromic curves.
Sharp proof of sub-Riemannian length-minimizing curves being at least C2
problem Smoothness of sub-Riemannian length-minimizing curves
method Study of a class of sub-Riemannian structures, proving C2 regularity result Theorem 1.1 in [6] is sharp
Convex curves evolve into circles over time.
problem Deforming convex curves into circles.
method Generalized length-preserving flow for convex curves.
result Convex curves evolve into circles over time.
The paper proves conditions for converting homotopies to monotone homotopies in Riemannian discs and spheres.
problem Conditions for converting homotopies to monotone homotopies in Riemannian discs and spheres.
method Analyzing the boundary of Riemannian discs and spheres to determine if they can be contracted monotonously.
result A monotone homotopy can be constructed for a Riemannian disc and sphere under certain length constraints.
Study compares hyperbolic and extremal lengths for shortest curves.
problem Comparing hyperbolic and extremal lengths for shortest curves.
method Lower bounds for widths of collars and upper bounds for renormalized volume of Schottky manifolds.
result Upper bounds of renormalized volume in terms of hyperbolic length of compressible curves.
Extends curve functions to geodesic currents with a simple criterion.
problem Continuous extension of curve functions to geodesic currents.
method Simple criterion based on smoothing property.
result Extends known curve functions and introduces new examples.
Global conformal parameters found for complex and hyperbolic curves.
problem Finding global conformal parameters for analytic curves.
method Analytic curves in complex and hyperbolic planes, and their generalizations.
result Spherical and hyperbolic arc-lengths are global conformal parameters for analytic curves.
The study counts curves on a once-punctured torus with self-intersections.
problem Counting closed curves with self-intersections on a once-punctured torus.
method Combinatorial classification of curves with given word-length and self-intersections.
result Determination of curve counts with zero, one, and arbitrary self-intersections.
Minimal translation lengths for Torelli and pure braid groups on curve graphs are shown.
problem Understanding translation lengths of Torelli and pure braid groups on curve graphs.
method Analyzing asymptotic translation lengths of Torelli and pure braid groups on curve graphs.
result Minimal asymptotic translation lengths for Torelli and pure braid groups on curve graphs are shown to behave differently from their respective mapping class groups.
Study approximate marked length spectrum rigidity in non-positively curved groups.
problem Approximate rigidity of marked length spectra in non-positively curved groups.
method Compare marked length spectra of isometric actions of groups with non-positively curved features.
result Supremum of quotient of marked length spectra is approximately determined by restricted spectra.
The study examines translation lengths of pseudo-Anosov maps on curve graphs.
problem Estimating translation lengths of pseudo-Anosov maps on curve graphs.
method Analyzing geodesic axes and powers of Dehn twists.
result Determining minimal translation lengths and optimizing map ratios.
Characterizes curves with short representatives on hyperbolic surfaces.
problem Inequalities on lengths of curves on hyperbolic surfaces.
method Characterization of topological types of curves and multicurves with short representatives.
result Characterizes which topological types of curves and multicurves always have a short representative.
New theorem shows certain curved surfaces are uniquely identified by their geodesic lengths.
problem Identifying surfaces by their geodesic lengths.
method Analyzes metrics on simple, thick negatively curved two-dimensional P-manifolds.
result Piecewise negatively curved Riemannian metrics on simple, thick two-dimensional P-manifolds are uniquely determined by their geodesic lengths.
Introduces fractional length and nonlocal curvature for smooth curves.
problem Defining curvature for curves of fractional length.
method Introduces fractional length and derives nonlocal curvature using fractional perimeter analogy.
result Fractional length converges to traditional length with a multiplicative constant.
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.
ReLU networks don't exponentially distort curve lengths as previously thought.
problem Understanding how neural networks distort curve lengths with depth.
method Analyzing expected length distortion of ReLU networks with random initialization.
result Expected length distortion does not grow with depth, and shrinks slightly.
Method approximates planar curves with circular arcs of equal length.
problem Approximating planar curves with circular arcs of equal length.
method Proposed by I.Kh. Sabitov and A.V. Slovesnov, extended with new inequalities and computer modeling.
result Derived inequalities for the length of a convex spiral arc with prescribed Hermite data.
This paper classifies and determines the length of the shortest filling pairs on a specific type of surface.
problem Classifying and determining the length of the shortest filling pairs on a specific type of surface.
method Classifying and determining the length of the shortest filling pairs on a specific type of surface.
result The paper classifies and determines the length of the shortest minimal filling pairs on a genus two surface.
Study finds minimum lengths of curves on a one-holed torus.
problem Minimizing the geodesic length of curves on a one-holed torus.
method Explicitly found minima and minimum points of geodesic length functions for a family of curves.
result Concrete examples provided for minimizing geodesic length on hyperbolic surfaces.