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.
The paper studies local normal forms of singular contact forms and primitive 1-forms.
problem Local normal forms of singular contact forms and primitive 1-forms.
method Combines classical normalization techniques and toric approach.
result Extends and improves previous results on first-order contact forms and primitive 1-forms.
Study of G2-structures with isolated singularities and bounded torsion.
problem Understanding G2-structures with special torsion and isolated singularities. method Revisiting known examples, describing symmetries, and analyzing collapsing of circle fibres.
result Collapsing circle fibres at isolated points cannot produce G2-structures with bounded torsion. Study on parabolic points and cylindrical surfaces in Euclidean 3-space.
problem Characterizing parabolic points and their geometric properties.
method Introducing contact cylindrical surfaces and analyzing their properties.
result Characterization of A-singularity through projections. The study of equidistants for families of surfaces, focusing on specific ratios of tangent planes.
problem Understanding the geometric properties of surfaces and their tangent planes.
method Local study of affine equidistants, critical value analysis of 2-parameter unfoldings, geometric classification of singularities.
result Classification of singularities in equidistants near the supercaustic chord.
We give necessary and sufficient geometric conditions for a distribution (or a Pfaffian system) to be locally equivalent to the canonical contact system on Jn(R,Rm), the space of n-jets of maps from R into Rm. We study the geometry of that class of systems, in particular, the existence of corank one involutive subdistr…
Rectifies flat singular points of area-minimizing currents with singularity degree > 1.
problem Rectifying flat singular points of area-minimizing currents with singularity degree > 1.
method Subdividing singular points based on singularity degree and proving rectifiability of points with singularity degree > 1.
result The set of points with singularity degree > 1 is (m-2)-rectifiable.
Investigate the local geometry of smooth surfaces in 4-space via contact with 2-planes and apparent contours.
problem Local geometry of smooth surfaces in 4-space
method Contact with 2-planes and apparent contour
result Prove connections between singularities of parallel projections, orthogonal projections, and height functions.
Unbraided wiring diagrams for Stein fillings of lens spaces are described.
problem Constructing Stein fillings of lens spaces with canonical contact structures.
method Algorithm to draw unbraided wiring diagrams equivalent to Lefschetz fibrations.
result Wiring diagrams can be extended to symplectic graphical disks with marked points.
Study Schrödinger evolution on surfaces in 3D contact sub-Riemannian manifolds.
problem Analyzing the Schrödinger evolution on surfaces embedded in 3D contact sub-Riemannian manifolds.
method Relating self-adjointness of the Schrödinger operator to geometric invariants of the foliation.
result Classification of self-adjoint extensions yielding disjoint dynamics.
Study on contact Hamiltonian functions for singular contact structures.
problem Understanding infinitesimal contact transformations on singular contact structures.
method Showed injectivity and provided an explicit local formula for the inverse map.
result Explicit local formula for the inverse map when contact structure has singularities of the first type.
New proof shows unique symplectic fillings for certain surface singularity links.
problem Uniqueness of symplectic fillings for specific rational surface singularity links.
method Analysis of positive monodromy factorizations for planar open books.
result Unique symplectic fillings proven for specified contact structures.
We study Legendrian singular links up to contact isotopy. Using a special property of the singular points, we define the singular connected sum of Legendrian singular links. This concept is a generalization of the connected sum and can be interpreted as a tangle replacement, which provides a way to classify Legendrian …
In this paper we provide the small-time heat kernel asymptotics at the cut locus in three relevant cases: generic low-dimensional Riemannian manifolds, generic 3D contact sub-Riemannian manifolds (close to the starting point) and generic 4D quasi-contact sub-Riemannian manifolds (close to a generic starting point). As …
Study on singularities of bundle homomorphisms induced by Morin maps.
problem Characterizing singular points of bundle homomorphisms.
method Analyzing conditions for singularities induced by Morin maps, using Hamilton vector fields for contact structures.
result Characterization of singularities in bundle homomorphisms induced by Morin maps.
Study on sub-Riemannian manifolds, focusing on conjugate points and caustic stability.
problem Analyzing sub-Riemannian manifolds and their conjugate points.
method Computed sub-Riemannian Hamiltonian flow, approximated conjugate locus, introduced geometric invariant.
result Contact distributions exhibit unique behavior, different from 3D case.
Study on contact structures of singularity links and existence of Stein cobordisms.
problem Existence problem of Stein cobordisms between contact structures of singularity links.
method Construction of explicit Stein cobordism and detection of contact Ozsvath-Szabo invariants.
result U-filtration depth obstructs the existence of Stein cobordism from proper almost rational to rational singularity.
Paper classifies singularities in 3D contact sub-Riemannian geometry.
problem Understanding singularities in sub-Riemannian geometry.
method Extended classification of singularities in 3D contact sub-Riemannian geometry.
result Complete classification of generic singularities.
Study shows infinite order in mapping class groups for certain 3D shapes.
problem Understanding the mapping class groups of certain 3D shapes.
method Analogues of Seiberg-Witten-Floer homology for 3-manifolds.
result Monodromy diffeomorphisms have infinite order in smooth mapping class groups.
Let L⊂J1(M) be a Legendrian submanifold of the 1-jet space of a Riemannian n-manifold M. A correspondence is established between rigid flow trees in M determined by L and boundary punctured rigid pseudo-holomorphic disks in T∗M, with boundary on the projection of L and asymptotic to the doubl…
Classifies real tight contact structures on lens spaces and solid tori.
problem Classifying real tight contact structures on specific 3-manifolds.
method Equivariant contact isotopy, real open book decompositions, and isolated real algebraic surface singularities.
result Unique real tight structures on S3 and RP3, at most one on L(p,±1), and bounds on the count. The regular type of a real hyper-surface M in an (almost) complex manifold at some point p is the maximal contact order at p of M with germs of non singular (pseudo) holomorphic disks. The main purpose of this paper is to give two intrinsic characterizations the type: one in terms of Lie brackets of a complex tangent v…
We establish a long exact sequence for Legendrian submanifolds L in P x R, where P is an exact symplectic manifold, which admit a Hamiltonian isotopy that displaces the projection of L off of itself. In this sequence, the singular homology H_* maps to linearized contact cohomology CH^* which maps to linearized contact …
The study proves that certain surface singularities are planar only for An-singularities.
problem Characterizing planar links of normal surface singularities.
method Using contact structures and symplectic fillings, the study applies obstructions to canonical contact structures.
result Links of isolated singularities of surfaces in the complex 3-space are planar only for An-singularities. Study on contact structures of surface singularity links using Heegaard Floer homology.
problem Understanding contact structures on singularity links using Heegaard Floer homology.
method Interplay between Heegaard Floer homology and Némethi's lattice cohomology.
result Heegaard Floer invariant cannot lie in the image of the U-action for canonical contact structures on singularity links.
Unified interpretation of sub-Riemannian Gauss-Bonnet theorem for surfaces in 3D contact manifolds.
problem Proving a sub-Riemannian Gauss-Bonnet theorem for surfaces in 3D contact manifolds.
method Measure-theoretic perspective, focusing on singular measures and characteristic points.
result Unified interpretation of previous results and natural geometric conditions for the theorem.
Given a manifold M and a proper sub-bundle Δ⊂TM, we study homotopy properties of the horizontal base-point free loop space Λ, i.e. the space of absolutely continuous maps γ:S1→M whose velocities are constrained to Δ (for example: legendrian knots in a contact manifold). A key technical ingredient f…
Study contact resolutions for Jacobi structures, providing examples and impossibility results.
problem Understanding contact resolutions of Jacobi structures.
method Examining various classes of Jacobi structures and their contact properties.
result Identified conditions under which contact resolutions exist and those where they do not.
Classifies and analyzes the stability of black hole event horizon birth points using contact geometry.
problem Classifying and understanding the structural possibilities of black hole crease sets.
method Contact geometry approach, focusing on BigFronts and their Legendrian projections.
result Refined stability discussion of the event horizon birth component and identification of additional components.
We study the equation for improper (parabolic) affine spheres from the view point of contact geometry and provide the generic classification of singularities appearing in geometric solutions to the equation as well as their duals. We also show the results for surfaces of constant Gaussian curvatureand for developable s…
In a recent paper of Akhmedov, Etnyre, Mark and Smith, it was shown that there exist infinitely many contact Seifert fibered 3-manifolds each of which admits infinitely many exotic (homeomorphic but pairwise non-diffeomorphic) simply-connected Stein fillings. Here we extend this result to a larger set of contact Seifer…
Two rigidity results for Legendrian singularities in complex-analytic category.
problem Understanding singularities of Legendrian subvarieties in contact manifolds.
method Using the relation between infinitesimal contactomorphisms and holomorphic sections of the natural line bundle.
result Normal Legendrian singularities are deformation-rigid.
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. This paper constructs a family of coordinate systems about a point on a quaternionic contact manifold, called quaternionic contact pseudohermitian normal coordinates. Once defined, conformal variations of the quaternionic contact structure induce changes on the coordinates which are studied in an effort to simplify the…
Cylindrical contact homology computed for links of simple singularities.
problem Computing cylindrical contact homology for links of simple singularities.
method Perturbing degenerate contact form on S3/G with an invariant Morse function, achieving nondegeneracy up to an action threshold. Recovers cylindrical contact homology via direct limit of action filtered homology groups. result Ranks of cylindrical contact homology groups are given in terms of ∣extConj(G)∣, demonstrating a form of the McKay correspondence. We show that the contact structure on the link of a cusp singularity is contactomorphic to a Sol-manifold with the positive contact structure arising from the Anosov flow.
New algebraic construction of knot contact homology using perverse sheaves.
problem Algebraic construction of knot contact homology.
method Defining a DG category with a braid group action on perverse sheaves.
result The endomorphism algebra of a distinguished object in the category matches fully noncommutative knot DGA.
We identify the canonical contact structure on the link of a simple elliptic or cusp singularity by drawing a Legendrian handlebody diagram of one of its Stein fillings. We also show that the canonical contact structure on the link of a numerically Gorenstein surface singularity is trivial considered as a real plane bu…
New points detected on surfaces in 4D space, revealing symmetries.
problem Detecting symmetries and special points on surfaces in 4D space.
method Contact classification of symmetric maps from the plane to the plane.
result Special parabolic points detected and associated with sign changes.
A 2-web in the plane is given by two everywhere transverse 1-foliations. In this paper we introduce the study of singular 2-webs, given by any two foliations, which may be tangent in some points. We show that such two foliations are tangent along a curve, which will be called the polar curve of the 2-web, and we study …
Study symplectic fillings of lens spaces, focusing on virtually overtwisted contact structures.
problem Classify symplectic fillings of virtually overtwisted contact structures on lens spaces.
method Use curve configurations on surfaces, algebraic properties of integer lattices, geometric slicing of solid tori, and connections to algebraic geometry.
result Find necessary conditions for Stein fillings to be Milnor fibers of hypersurface singularities.
We generalize the concept of locally symmetric spaces to parabolic contact structures. We show that symmetric normal parabolic contact structures are torsion--free and some types of them have to be locally flat. We prove that each symmetry given at a point with non--zero harmonic curvature is involutive. Finally we giv…
Planar multilinks prove rational singularities in surface geometry.
problem Characterizing surface singularities using planar multilinks.
method Combining topological and combinatorial approaches, including Min--Roy--Wang's work.
result Planar multilinks imply rational singularities and sandwiched singularities.
Study symplectic cohomology of certain singularities using homological mirror symmetry.
problem Compute symplectic cohomology for specific singularities.
method Use homological mirror symmetry to compute symplectic cohomology.
result Suggests a new conjecture about the relationship between small resolutions and symplectic cohomology.
We describe some natural relations connecting contact geometry, classical Monge-Ampere equations and theory of singularities of solutions to nonlinear PDEs. They reveal the hidden meaning of Monge-Ampere equations and sheds new light on some aspects of contact geometry.
Study the cuspidal edge's contact with planes and lines in 3D geometry.
problem Classify and understand the singularities of the cuspidal edge's contact with planes and lines.
method Classify submersions on a model of the cuspidal edge by diffeomorphisms, and use singularities of height functions and orthogonal projections to recover and describe the contact.
result Obtained generic singularities and deformations of the apparent contour, related height function and projection singularities to geometric invariants.
The paper provides an algorithm to create curves touching a smooth cubic at specific intersection points.
problem Creating curves that touch a smooth cubic at specific intersection points.
method Algorithm based on divisions and Zariski tuples to produce n-contact curves. result An algorithm to generate n-contact curves to a smooth cubic. The study explores conditions for realizing lens spaces as boundaries of Milnor fibers of hypersurface singularities.
problem Realizing other lens spaces as boundaries of Milnor fibers of hypersurface singularities.
method Examining necessary conditions for the realization of (L(p,q),ξ) as boundaries of Milnor fibers. result Obtained a series of necessary conditions for realizing other lens spaces as boundaries of Milnor fibers.