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.
For isolated complex hypersurface singularities with real defining equation we show the existence of a monodromy vector field such that complex conjugation intertwines the local monodromy diffeomorphism with its inverse. In particular, it follows that the geometric monodromy is the composition of the involution induced…
Paper describes links of mixed polynomials with specific properties.
problem Understanding the links of mixed polynomials with nice Newton boundaries.
method Analyzes links constructed from sequences of links associated with compact 1-faces of the Newton boundary.
result Links of singularities of inner non-degenerate mixed polynomials can be described using a specific procedure.
Using standard methods for studying singularities of projections and of contacts, we classify the stable singularities of affine λ-equidistants of n-dimensional closed submanifolds of Rq, for q≤2n, whenever (2n,q) is a pair of nice dimensions.
In L^3, cuspidal edges can have bounded mean curvature under specific conditions.
problem Understanding cuspidal edges with bounded mean curvature in Lorentz-Minkowski 3-space.
method Investigated cuspidal edges and generalized cuspidal edges, analyzing their singular points and principal curvatures.
result Cuspidal edges with bounded mean curvature in L^3 occur only when the singular set is a light-like curve.
A limaçon-like curve, allowing 2π-transition with monotone curvature between concentric curvature elements, is presented. The curve is 4th degree algebraic, 4th degree rational, and shares other common features with Pascal's limaçon.
We define winding numbers of regular closed curves on surfaces with a nice euclidean or hyperbolic geometry. We prove that two regular closed curves are regularly homotopic if and only if they are freely homotopic and have the same winding number.
The study shows manifolds with special generic maps also have nice multisections.
problem Characterizing manifolds with special generic maps and multisections.
method Analyzing manifolds with special generic maps and their properties, and showing how these maps restrict the differentiable structures of spheres and manifolds.
result Manifolds admitting special generic maps also admit nice generalized multisections.
We compute the algebraic equation of the universal family over the Kenyon-Smillie (2,3,4)-Teichmüller curve and give a nice geometric description of the torsion map. Moreover, we re-prove independently that the found algebraic equation describes a Teichmüller curve by computing the Picard-Fuchs equation associated to…
We prove that the Casimir operator acting on sections of a homogeneous vector bundle over a generalized flag manifold naturally extends to an invariant differential operator on arbitrary parabolic geometries. We study some properties of the resulting invariant operators and compute their action on various special types…
In this paper, we consider a CscK metric defined away from divisor and with metric upper bound and lower bound going to zero in certain rate. And we'll prove that this "nicely" behaved metric is a smooth CscK metric across the divisor.
Study on singularities of specific polynomial functions.
problem Characterizing the topology of singularities of mixed functions.
method Introduced inner non-degenerate mixed functions and used Newton boundary to characterize links.
result Links of singularities can be completely characterized under certain conditions.
A method is provided to resolve Lie algebroids with singularities.
problem Resolving Lie algebroids with singular points.
method Nash-type blow-up construction for Lie algebroids.
result A short exact sequence is established linking the blow-up to Lie algebroids.
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.
The paper characterizes timelike rectifying curves in De Sitter 3-space.
problem Characterizing timelike rectifying curves in De Sitter 3-space.
method Defining timelike rectifying curves and conical surfaces, providing characterizations and results.
result Characterizations and results of timelike rectifying curves in De Sitter 3-space.
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.
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.
We define a `nice representation' of a finitely presented group G as being a non-degenerate essentially surjective simplicial map f from a `nice' space X into a 3-complex associated to a presentation of G, with a strong control over the singularities of f, and such that X is WGSC (weakly geometrically simply connected)…
Study natural and conjugate mates of Frenet curves in Lie groups.
problem Characterize Frenet curves and their mates in Lie groups.
method Introduced natural and conjugate mates, derived relationships, analyzed specific curves.
result Obtained results for various Frenet curves in Lie groups.
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.
This paper deals with the question of J.Morava on existence of canonical complex cobordism class of singular submanifold. We present several solutions of this question for Xr(ξ) -- the set of points where dimξ−r+1 generic sections of a complex vector bundle ξ are linearly dependent. The corresponding complex co…
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.
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.
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 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.
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…
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 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.
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.
The study restricts manifolds with certain explicit SGL maps and constructs them.
problem Restrictions on manifolds admitting specific SGL maps.
method Generalization of Morse functions and canonical projections to construct SGL maps.
result Manifolds admitting certain explicit SGL maps are strongly topologically restricted.
A directed curve is a possibly singular curve with well-defined tangent lines along the curve. Then the tangent surface to a directed curve is naturally defined as the ruled surface by tangent geodesics to the curve, whenever any affine connection is endowed with the ambient space. In this paper the local diffeomorphis…
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.
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.
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 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…
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.
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.
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.