Developable surfaces near swallowtail singularities studied for flatness.
problem Understanding developable surfaces near swallowtail singularities.
method Introducing Darboux frames and invariants to study developable surfaces.
result New example of a frontal locally homeomorphic to a swallowtail.
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.
Swallowtails form on maxfaces converging to cone-like singularities.
problem Maxface convergence to cone-like singularities.
method Constructing maxface sequences with increasing swallowtails.
result Maxface sequences converge to cone-like singularities.
Study of flat geometry of swallowtail submersions.
problem Classifying submersions preserving the swallowtail.
method Classifying submersions from (R3,0) to (R,0) up to diffeomorphisms. result Derived flat geometry from contact with planes and singularities of height function.
The paper proves the existence of maxfaces with multiple swallowtails and planar ends.
problem Existence of maxfaces with specific geometric properties.
method Analyzes 1-parameter infinite genus families of maxfaces with swallowtails and planar ends.
result Existence of maxfaces with infinitely many swallowtails and planar ends.
The study characterizes surface singularities in Lie sphere geometry.
problem Understanding singularities of surfaces in Lie sphere geometry.
method Analyzing conditions for cuspidal edges, swallowtails, and Lie sphere transformations.
result Conditions for various surface singularities in Lie sphere geometry.
The paper describes the geometric properties of line congruences' singularities.
problem Understanding the singularities of generic line congruences.
method Use of an equiaffine pair to define generic line congruences.
result Geometric description of folds, cusps, and swallowtails as singularities of generic line congruences.
Timelike minimal surfaces in Lorentzian Heisenberg group have singular points.
problem Characterizing singularities on timelike minimal surfaces.
method Constructing timelike minimal surfaces as Lorentzian harmonic maps and analyzing their singularities.
result Criteria for cuspidal edges, swallowtails, and cuspidal cross caps are provided.
It is well-known that the unit cotangent bundle of any Riemannian manifold has a canonical contact structure. A surface in a Riemannian 3-manifold is called a (wave) front if it is the projection of a Legendrian immersion into the unit cotangent bundle. We shall give easily-computable criteria for a singular point on a…
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.
We prove that the difference between the numbers of positive swallowtails and negative swallowtails of the Blaschke normal map for a given convex surface in affine space is equal to the Euler number of the subset where the affine shape operator has negative determinant.
The study classifies singularities in discrete improper affine spheres.
problem Classifying singularities in discrete improper affine spheres.
method Analysis of discrete improper affine spheres based on asymptotic nets, distinguishing singular edges and vertices.
result First step in classifying singularities of discrete nets.
We shall give useful criteria of lips, beaks and swallowtail singularities of smooth map from the plane into the plane. As an application of criteria, we will discuss the singularities of Cauchy problem of single conservation law.
We investigate singularities of all parallel surfaces to a given regular surface. In generic context, the types of singularities of parallel surfaces are cuspidal edge, swallowtail, cuspidal lips, cuspidal beaks, cuspidal butterfly and 3-dimensional D4± singularities. We give criteria for these singularities type…
We define cuspidal curvature κc (resp. normalized cuspidal curvature μc) along cuspidal edges (resp. at swallowtail singularity) in Riemannian 3-manifolds, and show that it gives a coefficient of the divergent term of the mean curvature function. Moreover, we show that the product κΠ called the product curva…
We show that the singularities of spacelike maximal surfaces in Lorentz-Minkowski 3-space generically consist of cuspidal edges, swallowtails and cuspidal cross caps. The same result holds for spacelike mean curvature one surfaces in de Sitter 3-space. To prove these, we shall give a simple criterion for a given singul…
The study investigates deformations of swallowtails in 3D space, preserving curvature signs.
problem Deforming swallowtails in 3D space while maintaining curvature signs.
method Representation formula for swallowtails, investigation of map germs, and analysis of Gaussian curvatures.
result Swallowtails can be deformed into a swallowtail of constant Gaussian curvature while preserving curvature signs.
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…
We shall introduce the singular curvature function on cuspidal edges of surfaces, which is related to the Gauss-Bonnet formula and which characterizes the shape of cuspidal edges. Moreover, it is closely related to the behavior of the Gaussian curvature of a surface near cuspidal edges and swallowtails.
Study geometric singular solutions of generalized Monge-Ampère equations.
problem Solving generalized Monge-Ampère equations on a plane.
method Using exterior differential systems and Cauchy characteristics.
result Criteria for geometric singular solutions to be equivalent to specific types.
We use integrable systems techniques to study the singularities of timelike non-minimal constant mean curvature (CMC) surfaces in the Lorentz-Minkowski 3-space. The singularities arise at the boundary of the Birkhoff big cell of the loop group involved. We examine the behaviour of the surfaces at the big cell boundary,…
Study on stable maps from 3-sphere to 3-space, focusing on singularities and invariants.
problem Understanding the behavior of stable maps from S3 to R3. method Analysis of codimension-one transitions, singular set behavior, and global invariants.
result Effects of decompositions on global invariants with prescribed branch sets.
The paper explains how to realize Kossowski metrics as wave front germs.
problem Realizing Kossowski metrics as wave front germs at non-parabolic singular points.
method Using coherent tangent bundles and Kossowski's realization theorem.
result Criteria for realizing Kossowski metrics as induced metrics of singularities.
The paper studies singularities in discrete indefinite affine minimal surfaces.
problem Characterizing singularities in discrete indefinite affine minimal surfaces.
method Discretizing smooth curves and applying discrete Lelieuvre's formulas to study the resulting surfaces.
result The definition of singular edges and vertices in discrete asymptotic nets mirrors properties of smooth surfaces.
Extends Y.Eliashberg's h-principle to generic maps with prescribed Thom-Boardman singularities.
problem Homotoping maps with specific singularities.
method Proves a condition for continuous maps to be homotopic to generic maps with prescribed Thom-Boardman singularities.
result Necessary and sufficient condition for homotopy in dimension 3.
We define discrete flat surfaces in hyperbolic 3-space from the perspective of discrete integrable systems and prove properties that justify the definition. We show how these surfaces correspond to previously defined discrete constant mean curvature 1 surfaces in hyperbolic 3-space, and we also describe discrete focal …
In this paper, we introduce the notions of map-germs of pedal unfolding type and normalized Legendrian map-germs; and then we show that the fundamental theorem of calculus provides a natural one to one correspondence between Whitney umbrellas of pedal unfolding type and normalized swallowtails.
New maxfaces with catenoid or planar ends constructed using node-opening technique.
problem Lack of examples of maxfaces with catenoid or planar ends.
method Adapted node-opening technique to construct maxfaces of high genus.
result Singularities on constructed maxfaces form curves around the waists of the necks, with most singularities being cuspidal edges and the rest swallowtails.
In this paper, we give two classes of positive semi-definite metrics on 2-manifolds. The one is called a class of Kossowski metrics and the other is called a class of Whitney metrics: The pull-back metrics of wave fronts which admit only cuspidal edges and swallowtails in R3 are Kossowski metrics, and t…
Study of maximal surfaces in a specific Heisenberg group with singularities.
problem Characterize maximal surfaces in the Lorentzian Heisenberg group with singularities.
method Use harmonic maps into the 2-sphere, loop group construction, and solve the Cauchy problem.
result Regular maximal discs must have at least two cuspidal cross-cap singularities on the boundary.
The paper generalizes envelope constructions for chords in circles, revealing complex singularities.
problem Understanding envelopes of chords in circles with varying parameters and configurations.
method Generalizing the embroidery method to rational and concentric circles, breaking symmetry to reveal higher singularities.
result Higher singularities like swallowtails and butterflies can be unfolded, revealing their structure.
We study singularities of spacelike, constant (non-zero) mean curvature (CMC) surfaces in the Lorentz-Minkowski 3-space L3. We show how to solve the singular Björling problem for such surfaces, which is stated as follows: given a real analytic null-curve f0(x), and a real analytic null vector field v(x) paralle…
We show some generic (robust) properties of smooth surfaces immersed in the real 3-space (Euclidean, affine or projective), in the neighbourhood of a {\em godron} (term due to R.Thom): an isolated parabolic point at which the (unique) asymptotic direction is tangent to the parabolic curve. With the help of these proper…
Optimizes bounds for threefold singularity volumes.
problem Bounding local volumes of threefold singularities.
method Analyzes Gorenstein canonical non-hypersurface threefold singularities.
result Establishes optimal upper bound for local volumes.
Defines singular grid diagrams for various types of links.
problem Unified description of singular links and related objects.
method Definition of singular grid diagrams and classification of equivalence relations.
result Unified description of singular links and related objects.
Study describes singularities of height functions on specific singular surfaces.
problem Analyzing singularities of height functions on singular surfaces.
method Using geometric language and blowing-ups, investigate singularities of height functions and dual surfaces.
result Characterized singularities of height functions and dual surfaces on specific singular surfaces.
The paper extends affine connection results to singular warped and twisted products.
problem Generalizing affine connections to singular warped and twisted products.
method Study of singular multiply warped products and singular twisted products with semi-symmetric metric and non-metric connections, discussing Koszul forms and curvature.
result Theoretical results on curvature and Koszul forms for singular multiply warped and twisted products.
New singularity concept in GR: volume singularities.
problem Understanding spacetime singularities in General Relativity.
method Introducing a new type of singularity: volume singularities.
result Volume singularities are hidden by event horizons.
The study shows stability of neckpinch singularities in mean curvature flows.
problem Stability of neckpinch singularities in mean curvature flows.
method Analysis of mean curvature flow and perturbations.
result Stability of neckpinch singularities in mean curvature flows.
Study describes singularities of distance squared functions on singular surfaces.
problem Characterizing singularities of distance squared functions on singular surfaces.
method Using smooth map-germs Sk, Bk, Ck, and F4 singularities, the study describes singularities via blowing-ups. result Characterization of singularities of wave-fronts and caustics of singular surfaces.
Proves positive mass theorem for AF spin manifolds with conical singularities.
problem Proving the positive mass theorem for singular metrics on AF manifolds.
method Analyzes AF spin manifolds with isolated conical singularities, allowing topological singularities.
result Proves the positive mass theorem for AF spin manifolds with conical singularities.
The study examines singularities and geometric properties of surfaces derived from frontals with specific singular points.
problem Characterizing and understanding the singularities and geometric properties of surfaces formed by the singular loci of normal congruences of frontals with pure-frontal singular points.
method Characterizations of singularities in terms of geometric invariants of the initial frontal are provided for the normal ruled surface. Relations between certain singularities of focal surfaces and geometric properties of the frontal are also explored.
result Behavior of Gaussian curvature of focal surfaces of frontals with a 5/2-cuspidal edge is considered. Study singularities of Weinstein skeleta and arboreal singularities.
problem Understanding singularities in Weinstein skeleta and their relation to arboreal singularities.
method Deforming the skeleton via homotopies, Morse-Bott representative, generic perturbation, localization procedure.
result Singularities of Weinstein skeleta can be classified into arboreal singularities or tangency singularities.
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.
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.
Paper generalizes a theorem for real analytic singularities.
problem No specific problem stated; focuses on generalization.
method Generalization of a theorem for complex singularities.
result Generalized Join theorem for real analytic singularities.
The paper extends deformation theory to Calabi-Yau varieties with isolated log canonical singularities.
problem Deformation theory of Calabi-Yau varieties with log canonical singularities.
method Study of higher Du Bois and rational singularities, focusing on 0-liminal singularities.
result Existence of first order smoothings for isolated 0-liminal hypersurface singularities.
Study on singular twisted links and virtual braids, extending knot theory concepts.
problem Extending knot theory concepts to singular twisted links and virtual braids.
method Definition and analysis of singular twisted virtual braids and their monoid structure.
result Presentation of monoid and reduced monoid for singular twisted virtual braids.