Study introduces new indices for special fibered surfaces of genus 4.
problem Characterizing relatively minimal fibered surfaces of genus 4.
method Introduces Horikawa index and local signature for surfaces with unique trigonal structure.
result New indices provide insights into the structure of surfaces.
New group-theoretic Johnson classes applied to curves with torsion Ceresa classes.
problem Analyzing torsion in Ceresa classes of curves.
method Group-theoretic analogues of Johnson/Morita cocycles applied to pro-l etale fundamental groups of curves.
result Example of a non-hyperelliptic curve with torsion Ceresa class.
Analytic curves linked to algebraic ones via Schottky groups.
problem Moving between analytic and algebraic representations of Riemann surfaces.
method Identifying Riemann surfaces with Schottky groups and constructing families of non-hyperelliptic surfaces.
result Construction of families of non-hyperelliptic surfaces with specific properties.
For each d>=2, the mapping class group for plane curves of degree d will be defined and it is proved that there exists uniquely the Meyer function on this group. In the case of d=4, using our Meyer function, we can define the local signature for 4-dimensional fiber spaces whose general fibers are non-hyperelliptic comp…
The study characterizes a complex curve of residueless meromorphic differentials on elliptic curves.
problem Characterizing the locus of residueless meromorphic differentials on elliptic curves.
method Multi-scale compactification of strata, formulas for genus and degree of maps, distinguishing components.
result Complete classification of connected components of residueless loci in exceptional strata.
Study of trigonal curves in abelian differentials with specific divisor properties.
problem Characterizing locally closed subspaces of abelian differentials.
method Using linear systems on Segre-Hirzebruch surfaces to describe orbifold structure and orbifold fundamental groups.
result Identified the orbifold fundamental group of a specific subspace as a quotient of the Artin group of type E8. According to the work of Kontsevich-Zorich, the invariant that classifies non-hyperelliptic connected components of the moduli spaces of Abelian differentials with prescribed singularities,is the parity of the spin structure. We show that for the moduli space of quadratic differentials, the spin structure is constant o…
Researchers describe a Ceresa class for tropical and topological curves, linking algebraic and cohomological perspectives.
problem Explicitly describe the Ceresa class for non-hyperelliptic curves.
method Combining algebraic, tropical, and topological perspectives, defining the Ceresa class for curves and surfaces.
result The Ceresa class is torsion in all settings: tropical curves, topological surfaces, and smooth algebraic curves over C((t)). Study shows monodromy kernels are large, failing to prove commensurability in specific strata.
problem Proving commensurability of mapping class groups through monodromy kernels.
method Analyzing monodromy maps for specific strata in translation surfaces.
result Kernels of monodromy maps contain a non-abelian free group of rank 2.
The study finds a unique systole maximum in non-hyperelliptic surfaces.
problem Understanding systole functions on translation surfaces.
method Analyzing local and global maxima of systole functions.
result Local maxima are not global in non-hyperelliptic components.
We study the one parameter family of genus 2 Riemann surfaces defined by the orbit of the L-shaped translation surface tiled by three squares under the Teichmüller geodesic flow. These surfaces are real algebraic curves with three real components. We are interested in describing these surfaces by their period matrices.…
We study a secondary invariant, called the Meyer function, on the fundamental group of the complement of the dual variety of a smooth projective variety. This invariant have played an important role when studying the local signatures of fibered 4-manifolds from topological point of view. As an application of our study,…
Configurations of rigid collections of saddle connections are connected component invariants for strata of the moduli space of quadratic differentials. They have been classified for strata of Abelian differentials by Eskin, Masur and Zorich. Similar work for strata of quadratic differentials has been done in Masur and …
Classifies components of abelian differentials over Teichmüller space.
problem Classifying components of strata of abelian differentials.
method Computing monodromy groups and determining finite generating sets for r-spin stabilizer subgroups. result Complete classification of strata components for g≥5. Study confirms non-injective monodromy for even genus 4 translation surfaces.
problem Characterizing monodromy of translation surfaces in even genus 4.
method Analysis of orbifold classifying spaces and finite-type Artin groups.
result Monodromy of Heven(6) contains a non-abelian free group of rank 2. We construct new families of non-hyperelliptic Lefschetz fibrations by applying the daisy substitutions to the families of words (c1c2⋯c2g−1c2gc2g+12c2gc2g−1⋯c2c1)2=1, (c1c2⋯c2gc2g+1)2g+2=1, and (c1c2⋯c2g−1c2g)2(2g+1)=1 in the mapping …
New methods to construct curve pairs and their applications.
problem Constructing curve pairs and their properties.
method Using integral curves to study direction and donor curves.
result New methods to construct partner curves of unit speed curves.
New method to construct partner curves of non-lightlike curves.
problem Constructing partner curves for non-lightlike curves.
method Using integral curves in Minkowski 3-space, direction curve, and donor curve.
result New methods to construct partner curves of a unit speed non-lightlike curve.
The study explores Bertrand and Mannheim curves in 4D Euclidean space for framed curves.
problem Exploring Bertrand and Mannheim curves in 4D Euclidean space for framed curves.
method Defining and investigating Bertrand and Mannheim curves of framed curves in 4D Euclidean space.
result Bertrand and Mannheim curves exist even for framed curves in 4D Euclidean space, contrary to regular curves.
The study examines Bertrand Legendre curves in the unit tangent bundle over Euclidean plane.
problem Investigating properties of Legendre curves and their associated curves.
method Analyzing Bertrand Legendre curves and their associated curves, including parallel, evolute, and involute curves.
result Existence conditions and inverse operation for Bertrand Legendre curves are provided.
Method for generating new curves from plane curves on cylinders.
problem Generating new space curves from given plane curves.
method Defining a non-planar space curve on a right generalized cylinder and examining its focal curve.
result Parametric representation of the focal curve of a cylindrical curve.
The paper characterizes curves in pseudo-Galilean 4-space.
problem Characterizing curves in the pseudo-Galilean 4-space G14. method Investigation and characterisation of admissible curves in terms of curvature functions.
result Necessary and sufficient conditions for admissible rectifying curves in G14. The paper explores Bertrand and framed curves in 3D space.
problem Characterizing Bertrand and framed curves in Euclidean 3-space.
method Analyzing curves where tangent, normal, or binormal lines match another curve's lines.
result Conditions for the existence of Bertrand and framed curves are clarified.
The paper examines how closed curves on surfaces intersect and how this intersection determines the curves.
problem Determining closed curves on surfaces based on their intersections.
method Constructing and studying k-equivalent curves, analyzing intersections with other curves. result Curves are determined by their intersections with all other curves, but non-simple curves require infinitely many intersections to distinguish.
Approximating complex curves with simple parametric curves is widely used in CAGD, CG, and CNC. This paper presents an algorithm to compute a certified approximation to a given parametric space curve with cubic B-spline curves. By certified, we mean that the approximation can approximate the given curve to any given pr…
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.
Flow deforms locally convex curves to curves of constant k-order width.
problem Evolve locally convex curves to curves of constant k-order width.
method Introduced a nonlocal curvature flow to evolve locally convex curves in the plane.
result The flow converges to a smooth, locally convex curve of constant k-order width as time goes to infinity.
Modified curve shortening flow constructs λ-Angenent curve.
problem Constructing λ-Angenent curve. method Modified curve shortening flow
result Constructs λ-Angenent curve. Study on CR curves in 3-sphere, focusing on critical curves integration and existence.
problem Addressing the integration and existence of critical curves in the CR 3-sphere.
method Provided a procedure for the explicit integration of general critical curves and characterized closed curves.
result Existence of infinite countably many closed critical curves.
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.
Study rectifying curves in 3D multiplicative Euclidean space.
problem Investigate rectifying curves in a non-Newtonian geometry setting.
method Apply multiplicative differential-geometric concepts to rectifying curves.
result Classify multiplicative rectifying curves using spherical curves.
In classical curve theory, the geometry of a curve in three dimensions is essentially characterized by their invariants, curvature and torsion. When they are given, the problem of finding a corresponding curve is known as 'solving natural equations'. Explicit solutions are known only for a handful of curve classes, inc…
The paper characterizes pedal curves of quadratic curves.
problem Understanding pedal curves of quadratic curves.
method Analyzing the inverse construction of pedal curves.
result Characterization of pedal curves of quadratic curves.
Unified description of aesthetic curves through self-affinities.
problem Characterizing log-aesthetic curves and their properties.
method Reformulating and proving self-affinities of planar curves, integrating equiaffine geometry.
result Unified characterization of constant curvature curves in similarity and equiaffine geometries.
Primitive curves in handlebodies form a connected complex.
problem Understanding the structure of curves in handlebodies.
method Defining and analyzing primitive curves and constructing sequences between them.
result The primitive curve complex for a handlebody is connected.
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.
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.
Defines new curves from tangent indicatrix of curves, linking them to helices and slant helices.
problem Understanding and constructing helices and slant helices from spherical curves.
method Defining integral curves of Frenet vectors and using their curvatures.
result Established relationships and methods to create helices and slant helices from specific spherical curves.
In this study, we introduce a new type of surface curves called D-type curve. This curve is defined by the property that the unit Darboux vector W0 of a space curve r(s) and unit surface normal n along the curve r(s) satisfy the condition <n,W0>=constant. We point out that a D-type curve is a geodesic curve or an asymp…
Study on triharmonic curves in f-Kenmotsu manifolds.
problem Characterizing triharmonic curves in f-Kenmotsu manifolds.
method Investigation of necessary and sufficient conditions for Frenet curves, slant, and Legendre curves to be triharmonic. Proof of specific properties of triharmonic Frenet curves.
result Triharmonic Frenet curves with constant curvature are Frenet helices in three dimensional f-Kenmotsu manifolds.
The paper generalizes rectifying and normal curves in Lorentzian n-space.
problem Characterizing and classifying g−rectifying and g−normal curves in Lorentzian n-space. method Introducing a g−position vector field and defining g−rectifying and g−normal curves based on this field. result Comprehensive characterization and classification of g−rectifying and g−normal curves. New findings on hyperbolicity of fine curve graphs and their subgraphs.
problem Investigating hyperbolicity of fine curve graphs and their subgraphs.
method Analyzing large subgraphs of fine curve graphs and computing distances in specific cases.
result Large subgraphs of fine curve graphs contain flats of every finite dimension, indicating they are not hyperbolic.
Study of p-biharmonic curves and their properties.
problem Generalizing biharmonic curves to p-biharmonic curves. method Classification and analysis of p-biharmonic curves on surfaces and space forms. result Existence and stability of p-biharmonic curves on closed surfaces. In this paper we study null Bertrand curves in R14 under the assumption the curve has a Cartan frame. We show that if the derivative vectors of the null Cartan curve in R14 is linearly independent, then this curve is not a Bertrand curve. Since then the already known notion of null Bertrand curves in $R…
Compact curve solution emerges from non-compact curve.
problem Constructing solutions from non-compact curves.
method Slingshot solution to curve shortening flow.
result Compact embedded solution exists for a finite time.
Homotopy types of curve and arc complexes are studied.
problem Understanding the homotopy types of curve and arc complexes.
method Proving homotopy equivalence and contractibility of complexes.
result Fine curve complex is homotopy equivalent to curve complex, fine arc complex is contractible.
New curves generalize helix and rectifying curves.
problem Generalizing helix and rectifying curves.
method Introducing f-rectifying curves with f-position vector in rectifying plane.
result Classification and characterization of f-rectifying curves.
We classify curves in the moduli space of curves that are both Shimura- and Teichmueller curves: Except for the moduli space of genus one curves there is only a single such curve. We start with a Hodge-theoretic description of Shimura curves and of Teichmueller curves that reveals similarities and differences of the tw…