The paper studies singularities of pedal curves of hyperbolic frontals.
problem Investigating singularities of pedal curves of spacelike frontals in hyperbolic 2-space.
method Analyzing singularities of pedal curves based on dual curve germs and pedal point locations.
result The singularities of pedal curves depend on the singularities of the first hyperbolic Legendrian curvature germ and the pedal point for non-singular dual curve germs. For singular dual curve germs, additional dependence on both Legendrian curvature germs is observed.
Curvature criteria for A-simple singularities and their parallel curves identified.
problem Determining singularity types of A-simple singularities and their parallel curves.
method Defined curvature parameters and criteria for A-simple singularities.
result Criteria to determine singularity types of A-simple singularities and their parallel curves.
The study classifies singularities of spherical orthotomic curves.
problem Classifying singularities of spherical orthotomic curves.
method Defining spherical orthotomic curves and classifying their singularities.
result Singularities of spherical orthotomic curves are classified.
New definition of envelopes for curves in Lorentz-Minkowski plane, including singular points.
problem Vague classical definitions of envelopes for singular curves in Lorentz-Minkowski plane.
method Introduced a family of smooth curves with singular points, proposed a new definition of envelopes using lightlike tangential data, and examined contact at singular points.
result Properties of envelopes for curves with singular points in Lorentz-Minkowski plane.
This paper classifies singularities of tangent surfaces to directed curves.
problem Classifying singularities of tangent surfaces to directed curves.
method Local diffeomorphism classification of generic directed curves.
result Swallowtails and open swallowtails appear generically for tangent surface singularities.
Study isotopy of rational cuspidal curves in 4-manifolds.
problem Isotopy of rational cuspidal curves in 4-manifolds.
method Tame symplectic curves, pseudo-holomorphic curves, log pairs, 4-dimensional topology.
result Every rational cuspidal curve is isotopic to a complex curve in degrees up to 5.
The paper examines geometric invariants near a specific type of singular point.
problem The behavior of geometric invariants near a singular point of a surface or curve.
method Analysis of geometric invariants for surfaces and curves that are suspensions of singular curves.
result Evaluation of the orders of Gaussian and mean curvatures for the studied surfaces and curves.
Study curve shortening flow on Riemann surfaces with conical singularities.
problem Evolution of curves on Riemann surfaces with singular points.
method Curve shortening flow governed by a degenerate quasilinear parabolic equation.
result Evolving curves stay fixed at singular points and show collapsing and convergence results.
Research examines curves of degree 8 with specific singularities.
problem Existence of curves with prescribed singularities.
method Algebraic and symplectic approaches.
result Characterization of curves with specific singularities.
Study curve shortening flow on Riemann surfaces with conic singularities.
problem Analyzing curve shortening flow on surfaces with conic singularities.
method Generalized Huisken's comparison function to Riemann surfaces and surfaces with conic singularities. Reproofed Gage-Hamilton-Grayson theorem. Proved CSF can't touch conic singularities with cone angles ≤ π.
result CSF can't touch conic singularities with cone angles ≤ π for embedded simple closed curves.
The Whitham flow for hyperelliptic curves has singularities that can be smoothly extended.
problem Singularities in the Whitham flow for hyperelliptic spectral curves.
method Analysis of deformations preserving periods of a meromorphic differential.
result Stable and unstable manifolds are non-empty, and the flow can be extended through the singularity.
Study on curve shortening flow with Neumann boundary conditions, showing singularities and limits.
problem Analyzing singularities in curve shortening flow with Neumann boundary conditions.
method Criterion for initial curves leading to singularities, proof of singularity type, rescaling analysis.
result Existence of type II singularities and limit shapes for convex initial curves.
Curve Shortening Flow preserves circularity for convex projections.
problem Understanding the behavior of curves under Curve Shortening Flow.
method Contradiction argument and analysis of tangent flows.
result Smooth curves with convex projections become asymptotically circular under Curve Shortening Flow.
The paper provides criteria and curvatures for singularities of curves in R^N.
problem Tackles the classification and characterization of singularities of curves in R^N.
method Systematic procedure for constructing criteria, explicit criteria for multiplicities 2-4, generalized curvatures.
result Generalized curvatures reinterpret Fukui's theorem for curves of finite multiplicities.
Paper proves certain curves don't form Type II singularities.
problem Proving non-existence of Type II singularities for specific curves.
method Minimal surface theory and isoperimetric ratio analysis.
result Proves certain curves don't form Type II singularities before collapsing.
The study examines singularities of ruled surfaces using RM vectors.
problem Analyzing singularities of ruled surfaces.
method Using Legendre curves and rotation minimizing frames.
result Classification of singularities of ruled surfaces.
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.
Study of singular curves in a specific type of hyperbolic distribution.
problem Characterizing singular curves in hyperbolic (4,7)-distributions. method Introduced hyperbolic (4,7)-distributions of type C3, described singular curves via prolongations. result Completely described singular curves for hyperbolic (4,7)-distributions of type C3. Study links curve singularities to quiver mutations.
problem Understanding the relationship between curve singularities and quiver mutations.
method Investigates the connection between the topology of curve singularities and the mutation equivalence of quivers associated with their morsifications.
result Established a connection between the topology of isolated curve singularities and the mutation equivalence of quivers.
Study null surfaces of pseudo-spherical curves in anti-de Sitter space.
problem Characterize null surfaces of pseudo-spherical spacelike framed curves in anti-de Sitter 3-space.
method Introduced nullcone fronts, classified singularities, defined Anti-de Sitter distance-squared functions.
result Relate singularities of nullcone fronts to those of framed curves.
Study of singularities in a modified curve shortening flow with density.
problem Understanding singularities in the curve shortening flow with a density.
method Define and analyze Type I singularities, describe blow-up behavior, and study limits of blow-ups.
result Characterized a family of curves producing only Type I singularities under the modified flow.
Classifies singularities in tangent surfaces of generic curves.
problem Classifying singularities in tangent surfaces of generic space curves.
method Local diffeomorphism classification using tangent geodesics and frontal singularities.
result Complete solution to the local diffeomorphism classification problem.
The paper examines singular points in Wigner caustics and affine equidistants of planar curves.
problem Analyzing singular points in Wigner caustics and affine equidistants of planar curves.
method Generalizing the Blaschke-Süss theorem to study convex curves and their antipodal pairs.
result Existence of antipodal pairs in convex curves is generalized.
We study a generalization of constant Gauss curvature -1 surfaces in Euclidean 3-space, based on Lorentzian harmonic maps, that we call pseudospherical frontals. We analyze the singularities of these surfaces, dividing them into those of characteristic and non-characteristic type. We give methods for constructing all n…
Study helicoidal surfaces with singular points using frontals.
problem Investigate helicoidal surfaces with singular points.
method Use frontals in the Euclidean plane to analyze helicoidal surfaces.
result Provide criteria for the singularities of helicoidal surfaces of frontals.
In this paper, we study the computation of curvatures at the singular points of algebraic curves and surfaces. The idea is to convert the problem to compute the curvatures of the corresponding regular parametric curves and surfaces, which have intersections with the original curves and surfaces at the singular points. …
Study of curve shortening flow for twisted curves, defining curvature-torsion entropy.
problem Understanding the behavior of curves with curvature and torsion under curve shortening flow.
method Defined curvature-torsion entropy to analyze the flow of twisted curves.
result Curved curves under curve shortening flow either develop inflection points or exhibit highly irregular singularities.
The geometric monodromy of a plane curve singularity is a quasi-finite diffeomorphism. In this paper we locate the reduction curves of the geometric monodromy and the quadratic vanishing cycles of the singularity. An application to the geometric monodromy group is given.
We provide a condition for spatial curves which rules out the development of a type I singularity. The condition is that after the last time for which an inflection point develops, if the torsion is ever everywhere non-negative, the curve cannot develop a type I singularity.
Study singular curves on subriemannian spaces that don't affect homotopy types.
problem Understanding how singular curves affect the topology of horizontal paths.
method Analyzing the subriemannian energy and endpoint map to identify homotopically invisible curves.
result For d≥3, generic subriemannian structures have only homotopically invisible singular curves. New Stein fillings found for rational surface singularities.
problem Exploring Stein fillings of rational surface singularities.
method Using planar open books and Lefschetz fibrations, describe Stein fillings via symplectic disk arrangements.
result Many rational singularities admit Stein fillings not diffeomorphic to Milnor fibers.
The paper studies singularities on parallels of tangent developable surfaces of frontal curves.
problem Understanding singularities on parallels of tangent developable surfaces.
method Generalization of tangent developable surfaces and parallel deformations for frontal curves in arbitrary dimensions.
result Classification of generic singularities on parallels of tangent developable surfaces for frontal curves in 3 or 4 dimensional Euclidean spaces.
This paper proves geodesic curvature measures are bounded for curves near cross cap singularities.
problem Boundedness of geodesic curvature measures near cross cap singularities.
method Analyzes intrinsic cross cap singularities and extends Gauss-Bonnet formula.
result Proves boundedness of geodesic curvature measures for curves near cross cap singularities.
The Abstract Boundary singularity theorem was first proven by Ashley and Scott. It links the existence of incomplete causal geodesics in strongly causal, maximally extended spacetimes to the existence of Abstract Boundary essential singularities, i.e., non-removable singular boundary points. We give two generalizations…
Study on focal surfaces and evolutes of framed curves in hyperbolic 3-space using Legendrian duality.
problem Investigate differential geometry properties of framed curves, including singular points.
method Use Legendrian dualities to analyze focal surfaces and evolutes of hyperbolic framed curves.
result Show the relationship among focal surfaces, evolutes, and dual surfaces of evolutes.
The paper studies the structure of endpoint maps near nice singular curves in sub-Riemannian structures.
problem Understanding the structure of endpoint maps near nice singular curves in sub-Riemannian structures.
method Finding a normal form for the endpoint map and studying its restriction to level sets of the action functional.
result The endpoint map can be written as a sum of a linear map and a quadratic form locally around a nice singular curve.
Defines distinguished curves for Poincaré-Einstein and singular geometries.
problem Characterize distinguished curves for Poincaré-Einstein and singular geometries.
method Characterizes curves agreeing with geodesics away from singularities and satisfies boundary conditions.
result Provides a general theory of first integrals for distinguished curves in (Poincaré-)Einstein manifolds.
Invariants count inflections and vertices in singular plane curves.
problem Counting inflections and vertices in singular plane curves.
method Defining invariants If and Vf to count inflections and vertices, respectively, and analyzing their properties. result The invariants If and Vf are finite and bounded for curves without smooth components. Proves Goh conditions for singular curves with specific properties.
problem Finding optimal paths with specific geometric constraints.
method Proof of Goh conditions of order n and open mapping theorem.
result Establishes conditions for strictly singular curves of corank 1.
Study on singularities of frontal surfaces, classifying under equivalence.
problem Classifying singularities of frontal surfaces.
method Classification under left-right-equivalence, introduction of frontalisation, definition of cuspidal and transverse double point curves.
result Frontal surfaces have finite codimension if and only if the curves are reduced.
Constructs Lagrangian skeleta for curve singularities.
problem Understanding Lagrangian skeleta of curve singularities.
method Constructs closed arboreal Lagrangian skeleta associated to links of isolated plane curve singularities.
result Provides computations of Legendrian and Weinstein invariants.
Method counts connected 2D stratifolds with singular curves and components.
problem Counting 1-connected trivalent 2-stratifolds. method Describes a method for counting.
result Counts 1-connected trivalent 2-stratifolds. Study constraints on singular points of rational cuspidal curves using Heegaard Floer theory.
problem Constraints on singular points of rational cuspidal curves.
method Involutive Heegaard Floer homology theory.
result Results do not apply to rational cuspidal curves of even degree.
Curve shortening flow converges to a point with entropy bound.
problem Analyzing the behavior of curves under shortening flow near singularities.
method Analyzes blow-up limits and uses entropy bounds to prove convergence.
result Initial curves with entropy bound converge to a round point in finite time.
The paper extends a geometric model using singular curves.
problem Understanding abnormal extremals in sub-Riemannian geometry.
method Analysis of singular curves and construction of a graded Lie algebra.
result A nilpotent graded Lie algebra is constructed isomorphic to F4. Divides help construct fibered links from singularities.
problem Understanding complex isolated plane curve singularities.
method Using divides to topologically construct fibered links.
result Explicitly given monodromy diffeomorphism as a product of Dehn twists.
Study on Bertrand lightcone framed curves in Lorentz-Minkowski 3-space.
problem Analyzing mixed types of curves with singular points in Lorentz-Minkowski 3-space.
method Using lightcone frame to consider Bertrand types for lightcone framed curves.
result Existence conditions of Bertrand lightcone framed curves in all cases.
New proof shows existence of maximal surfaces with special singularities.
problem Existence of maximal surfaces with specific singularities.
method Different formulation and proof for singular Björling problem.
result Existence of maximal surfaces containing a given curve with special singularities.