Curve shortening flow converges to a translating soliton for certain curves.
problem Asymptotic behavior of curve shortening flow.
method α-curve shortening flow for exponents α > 1/2.
result Converges to the unique translating soliton.
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48 results for “asymptotic curve”
Study asymptotics of special curves on shrinking cylinders.
problem Asymptotic behavior of holomorphic curves on degenerating cylinders.
method Analysis of holomorphic curves in symplectizations with harmonic perturbation.
result Established precise asymptotic formulas for small area curves.
Study non-fillable curves in a hyperbolic surface with a real line.
problem Non-fillable curves in
method Asymptotic Plateau problem in
result First examples of non-fillable finite curves with no thin tail.
Minimal asymptotic translation lengths of Torelli groups and pure braid groups on the curve graphmath.GT
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.
The paper studies a flow of Legendre curves, generalizing the inverse curvature flow of regular curves.
problem Analyzing the inverse curvature flow of Legendre curves.
method Investigates the unique existence, monotonicity, and asymptotic behavior of the flow.
result The flow asymptotically converges to a self-similar solution, categorized by initial curve.
We analyzed the problem of finding a surfaces family through an asymptotic curve with Cartan frame. We obtain the parametric representation for surfaces family whose members have the same as an asymptotic curve. By using the Cartan frame of the given null curve, we present the surface as a linear combination of this fr…
Two curves in hyperbolic space share a bounded distance, leading to a minimizing surface.
problem Finding a surface minimizing area between two disjoint curves in hyperbolic space.
method Analyzing the asymptotic boundary of hyperbolic 3-space, applying Definition 1.8 for distance bounds, and proving Theorems 1.7 and 1.11.
result Existence of an area-minimizing surface between two disjoint curves with bounded distance.
In the present paper, we handle the problem of finding a hypersurface family from a given asymptotic curve in R^4. Using the Frenet frame of the given asymptotic curve, we express the hypersurface as a linear combination of this frame and analyze the necessary and sufficient conditions for that curve to be asymptotic. …
Surfaces and curves play an important role in geometric design. In recent years, problem of finding a surface passing through a given curve has attracted much interest. In the present paper, we propose a new method to construct a surface interpolating a given curve as the asymptotic curve of it. Also, we analyze the co…
Finite type curves are asymptotic lines of plane fields, with examples.
problem Characterizing finite type curves as asymptotic lines of plane fields.
method Proving finite type curves are asymptotic lines of suitable plane fields, providing examples.
result Generalizations of V. Arnold's results for plane fields.
Study on frequencies of non-simple curves in surfaces of large genus.
problem Frequency of non-simple curves in surfaces of large genus.
method Expression for frequency, large genus asymptotics, comparison with previous work.
result Identify most common types of non-simple curves with K intersections.
We find an upper bound for the asymptotic dimension of a hyperbolic metric space with a set of geodesics satisfying a certain boundedness condition studied by Bowditch. The primary example is a collection of tight geodesics on the curve graph of a compact orientable surface. We use this to conclude that a curve graph h…
Study shows Betti numbers of curves and orbifolds relate to volume and genus.
problem Understanding Betti numbers of Shimura curves and 3-orbifolds.
method Analyzes asymptotic behavior of Betti numbers in relation to volume and genus.
result Gauss-Bonnet equality for Shimura curves and vanishing Betti numbers for 3-orbifolds.
Corrected an error in yield curve behavior models.
problem Error in the boundary expression for yield curve shapes.
method Revised the mathematical expression for yield curve behavior.
result Corrected the boundary expression for normal and humped yield curves.
Solves area-minimizing surface problem for finite curves in H^2xR.
problem Asymptotic Plateau problem for area-minimizing surfaces.
method Complete solution for finite curves in $\BHH$.
result Fairly complete solution for finite curves in $\BHH$.
We simplify swallowtail singularity and classify nearby curves.
problem Understanding swallowtail singularity and its nearby curves.
method Coordinate transformation on source and isometry on target.
result Classification of asymptotic and characteristic curves near swallowtail.
In this paper we prove a universal inequality describing the asymptotic behavior of support points for planar continuous curves. As corollaries we get an analogous result for tangent points of differentiable planar curves and some (partially known) assertions on the asymptotic of the mean value points for various class…
We provide a detailed description of solutions of Curve Shortening in that are invariant under some one-parameter symmetry group of the equation, paying particular attention to geometric properties of the curves, and the asymptotic properties of their ends. We find generalized helices, and a connection with curv…
Characterizes rotational solitons for curve shortening flow on revolution surfaces.
problem Understanding the behavior of curves under curve shortening flow on revolution surfaces.
method Characterization and asymptotic behavior analysis.
result Asymptotic behavior of rotational solitons to parallel geodesics.
In this paper, we express surfaces parametrically through a given spacelike (timelike) asymptotic curve using the Frenet frame of the curve in Minkowski 3-space. Necessary and sufficient conditions for the coefficients of the Frenet frame to satisfy both parametric and asymptotic requirements are derived. We also prese…
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.
Solves asymptotic -realization problem for curves.
problem Realization problem for plane curves.
method Asymptotic analysis of smooth -realization.
result Determines cobordism distance between specific knot types.
Mirzakhani extended curve counting from simple to all curves.
problem Counting curves in the mapping class group orbit with length at most L.
method Low-tech argument showing general result from simple curves.
result Derived the asymptotic growth of curves for arbitrary curves.
Study on minimal submanifolds in curved spaces with unique solution to asymptotic Plateau problem.
problem Minimal submanifolds in negatively curved spaces with small curvature.
method Analysis of spheres at infinity and asymptotic Plateau problem.
result Complete minimal submanifolds bound a class of spheres with uniquely solvable asymptotic Plateau problem.
Study on closed curves on negatively curved surfaces, linking number formula, and restrictions.
problem Isometric rigidity of tight surfaces and properties of closed asymptotic curves.
method Using Călugăreanu's theorem, derive a formula for the linking number and analyze properties of curves.
result Closed curves with zero linking number cannot have certain planar projections.
The study examines pseudo-Anosov maps on curve complexes and their asymptotic translation lengths.
problem Analyzing the behavior of pseudo-Anosov maps on curve complexes.
method Using sequences of fibers and monodromies in the fibered cone, the asymptotic translation length of pseudo-Anosov maps is studied.
result The asymptotic translation length of pseudo-Anosov maps on curve complexes behaves asymptotically like .
Metrics are semipositively curved if they meet a specific asymptotic condition.
problem Characterizing semipositively curved metrics in Hermitian geometry.
method Proving metrics are semipositively curved if and only if they satisfy an asymptotic extension property.
result Proves a specific condition for Griffiths semipositively curved metrics.
We prove by Hilbert-Mumford criterion that a slope stable polarized weighted pointed nodal curve is Chow asymptotic stable. This generalizes the result of Caporaso on stability of polarized nodal curves, and of Hasset on weighted pointed stable curves polarized by the weighted dualizing sheaves. It also solved a questi…
Study on helix curves and their Möbius energy asymptotics.
problem Understanding the asymptotic behavior of Möbius energy for helix curves.
method Investigation of helix curves with fixed radius, focusing on energy decay and blow-up.
result Proven asymptotics for both uncoiling and coiling helix curves, revealing distinct strategies for each.
Polynomial-time algorithm for curve graph geodesics.
problem Computing distances and properties of mapping classes.
method Polynomial-time algorithm for geodesics in curve graphs.
result Asymptotic translation length, Nielsen-Thurston type, and canonical curve system can be computed in polynomial time.
We prove that a class of asymptotically nonnegatively curved manifolds (in the sense of Abresch) satisfying some uniform Euclidean type volume growth conditions contains only finitely many homeomorphism types.
The grand arc graph's asymptotic dimension is shown to be infinite.
problem Determining the asymptotic dimension of the grand arc graph.
method Using Gromov-hyperbolic and cocompact arc and curve models, the asymptotic dimension is shown to be infinite for a broad class of surfaces.
result The asymptotic dimension of the grand arc graph is infinite.
We give a bound, linear in the complexity of the surface, on the asymptotic dimension of the curve complex as well as the capacity dimension of the ending lamination space.
Study minimal translation lengths on curve complexes, providing bounds and constructing examples.
problem Understanding minimal translation lengths on curve complexes and their relation to Teichmüller spaces.
method Analysed the curve complex analog of Teichmüller spaces, providing lower and upper bounds and constructing specific examples.
result Lower bound on minimal asymptotic translation length on curve complexes interpolates known results on Teichmüller spaces.
Negative-curvature surfaces uniquely define boundary distances.
problem Defining distances on curved surfaces at infinity.
method Renormalized distance calculation for negatively-curved surfaces.
result Negative-curvature surfaces are boundary distance rigid.
Study of height jumps in Ceresa cycle using asymptotic Hodge theory.
problem Understanding height jumps in the Ceresa cycle.
method Analysis of asymptotic behavior of Hain-Reed beta-invariant in degenerating families of curves.
result Height jump of Ceresa cycle is equal to the slope of the dual graph of the curve.
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.
Upper bounds on translation lengths in fibered 3-manifolds.
problem Bounding translation lengths in fibered 3-manifolds.
method Analyzing asymptotic behavior of pseudo-Anosov monodromies in fibered cones.
result Upper bounds on asymptotic translation lengths derived from fibered faces.
In this paper, we analyze the asymptotic behavior of -noncollapsed and positively curved steady Ricci solitons and prove that any -dimensional -noncollapsed steady Kähler-Ricci soliton with non-negative sectional curvature must be flat.
The paper examines the geometry of a curve's centre symmetry set.
problem Global geometrical properties of a curve's centre symmetry set.
method Study of the envelope of affine chords.
result Number of singularities and asymptotes of the centre symmetry set.
Study curves evolving under curvature motion in networks.
problem Existence and behavior of network curves under curvature motion.
method Existence, uniqueness, singularity formation, asymptotic behavior of network curves.
result Understanding of network curves evolving under curvature motion.
Curves become nearly circular over time without initial assumptions.
problem Understanding the asymptotic behavior of area-preserving flows.
method Proving asymptotic circularity without additional assumptions.
result Immortal solutions become asymptotically circular without initial assumptions.
Study mapping class group orbits of curves with self-intersections on surfaces.
problem Counting and understanding orbits of curves with self-intersections on surfaces.
method Counting embeddings of ribbon graphs, asymptotic analysis, and counting curves with fixed minimal genus.
result Asymptotic number of orbits of curves with bounded self-intersections and minimal genus.
On properties of the sets of positively curved Riemannian metrics on generalized Wallach spacesmath.DG
Curved metrics on Wallach spaces bounded by curves under flow.
problem Properties of positively curved Riemannian metrics on Wallach spaces.
method Normalized Ricci flow analysis on specific Wallach spaces.
result Set of metrics forms bounded curves asymptotically.
Compact, non-convex curve flows are created.
problem Creating compact, non-convex ancient solutions for curve shortening flow.
method Constructed an ancient solution asymptotic to Yin-Yang curve.
result Compact, non-convex ancient solutions for curve shortening flow are demonstrated.
The paper examines asymptotic lines of plane fields in 3D space.
problem Qualitative properties of asymptotic lines in plane fields.
method Analysis of null directions and Gaussian curvature.
result Asymptotic lines coincide with classical ones in completely integrable fields.
Counting meanders on surfaces of arbitrary genus, with precise asymptotics.
problem Counting and understanding meanders on surfaces of arbitrary genus.
method Square-tiled surfaces, moduli spaces of Abelian and quadratic differentials, Witten-Kontsevich 2-correlators.
result Asymptotic probability and polynomial growth of meanders with intersections.
Study introduces D-type curves on surface pencils and their properties.
problem Characterizing curves on surface pencils.
method Using Frenet vectors and parametric representation, investigate necessary and sufficient conditions for curves to be D-type curves.
result D-type curves are geodesic or asymptotic in some cases.