Study symplectic invariants of parabolic orbits and cuspidal tori in integrable systems.
problem Understanding symplectic invariants of degenerate singularities in integrable systems.
method Normal forms and new techniques for studying symplectic invariants.
result New insights into symplectic invariants of degenerate singularities.
Researchers introduce new invariants for a specific type of edge geometry.
problem Investigating geometric properties of 5/2-cuspidal edges. method Introducing secondary cuspidal curvature and bias, proving their properties and product's invariance.
result Real analytic 5/2-cuspidal edges with non-vanishing limiting normal curvature admit non-trivial isometric deformations. Extends duality preserving singular set images and first fundamental forms to generalized cuspidal edges.
problem Preserving singular set images and first fundamental forms on generalized cuspidal edges.
method Extends previous isometric duality to generalized cuspidal edges including cuspidal cross caps and 5/2-cuspidal edges.
result New geometric insights on the duality.
Study on cuspidal edges and cross-caps in 3D geometry.
problem Understanding the geometry of folded cuspidal edges and cross-caps.
method Analyzing geometrical invariants and submersions preserving flat geometry.
result Geometrical invariants uniquely determine cuspidal cross-caps up to order 5.
We study parallel surfaces and dual surfaces of cuspidal edges. We give concrete forms of principal curvature and principal direction for cuspidal edges. Moreover, we define ridge points for cuspidal edges by using those. We clarify relations between singularities of parallel and dual surfaces and differential geometri…
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.
Along cuspidal edge singularities on a given surface in Euclidean 3-space, which can be parametrized by a regular space curve, a unit normal vector field ν is well-defined as a smooth vector field of the surface. A cuspidal edge singular point is called generic if the osculating plane of the cuspidal edge (as a regul…
Study geometric properties of cuspidal edges with boundary.
problem Differential geometric properties of cuspidal edges with boundary.
method Analysis of differential geometric invariants and their relations.
result Relation between boundary behavior and other invariants.
Study of cuspidal edges on focal surfaces of regular surfaces.
problem Clarifying the sign of singular curvature at cuspidal edges.
method Investigation using singularities of parallel surfaces.
result Clarification of the sign of singular curvature at cuspidal edges.
Study relates Gaussian curvature signs to cuspidal edge types and geometric invariants.
problem Understanding the relationship between Gaussian curvature and singularities of Gauss maps of cuspidal edges.
method Analyzes geometric invariants and types of singularities of Gauss maps to define and characterize positivity/negativity of cusps.
result Defines and characterizes positivity/negativity of cusps of Gauss maps by geometric invariants of cuspidal edges, and shows relation between sign of cusps and Gaussian curvature.
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…
Proves existence and uniqueness of CMC foliations in cuspidal manifolds.
problem Existence and uniqueness of CMC foliations in asymptotically cuspidal manifolds.
method Proof without curvature assumptions, applicable in any dimension.
result Proves existence and uniqueness of CMC foliations.
We give useful criteria for S_1 singularities in the Mond classification table, and cuspidal S_k singularities. As applications, we give a simple proof of a result given by Mond and a characterization of cuspidal S_k singularities for the composition of a cuspidal edge and a fold map indicated by Arnol'd for the case k…
Study dualities of geometric invariants on cuspidal edges in hyperbolic and de Sitter spaces.
problem Computing and understanding dualities of geometric invariants on cuspidal edges.
method Analyzing differential geometric invariants of cuspidal edges in hyperbolic and de Sitter spaces.
result Identified dualities of invariants on cuspidal edges.
Geometric study of cuspidal S1 singularities using diffeomorphisms and isometries.
problem Understanding geometric properties of cuspidal S1 singularities. method Form representing deformation using diffeomorphisms and isometries, necessary and sufficient condition for frontal maps.
result Investigation of geometric properties and cuspidal cross caps in deformations.
Research examines geometric foliations on cuspidal edges.
problem Understanding the geometric configurations of curvature lines on cuspidal edges.
method Analyzes 3-jets of parametrizations to determine configurations.
result Identifies key topological configurations of curvature lines.
Paper studies singularities of timelike minimal surfaces in Minkowski 3-space.
problem Exploring singularities of timelike minimal surfaces in Minkowski 3-space.
method Existence and non-existence theorems, criteria for specific singularities.
result Various singularities unique to timelike minimal surfaces, including cuspidal butterfly and (2,5)-cuspidal edge. Maxfaces can have cuspidal edges near certain singularities.
problem Characterizing singularities on maxfaces.
method Analyzing singular Björling data and proving geometric properties.
result Near a maxface with a specific type of singularity, there exists another maxface with a cuspidal edge.
The paper finds infinitely many cuspidal edges along a knot with the same first fundamental form.
problem Finding cuspidal edges with the same first fundamental form along a knot.
method Analyzes Cω-cuspidal edges along a knot C. result Infinitely many non-congruent cuspidal edges have the same first fundamental form.
The paper studies parallel surfaces of cuspidal cross caps and their degeneracy.
problem Investigating the geometry and singularities of parallel surfaces of cuspidal cross caps.
method Established a criterion for the degeneracy of the distance squared function using geometric invariants.
result Parallel surfaces degenerate into a degenerated cuspidal S1 singularity at specific distances.
We give a normal form of the cuspidal edge which uses only diffeomorphisms on the source and isometries on the target. Using this normal form, we study differential geometric invariants of cuspidal edges which determine them up to order three. We also clarify relations between these invariants.
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.
The study of symplectic fillings for rational cuspidal curves.
problem Understanding symplectic fillings of contact manifolds associated with rational cuspidal curves.
method Exploration through Stein handlebodies and rational blow-downs.
result Examples of contact manifolds that are links of normal surface singularities, and those that do not admit symplectic fillings.
Classifies curves up to symplectic isotopy.
problem Classifying rational cuspidal curves up to symplectic isotopy.
method Topological tools, pseudoholomorphic techniques, and birational transformations.
result Classifies rational cuspidal curves of degrees 6 and 7 up to symplectic isotopy.
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 use invariants of Hendricks and Manolescu coming from involutive Heegaard Floer theory to find constraints on possible configurations of singular points of a rational cuspidal curve of odd degree in the projective plane. We show that the results do not carry over to rational cuspidal curves of even degree.
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…
Study maxfaces and minfaces converging to surfaces with folded singularities.
problem Analyzing surfaces with folded singularities and their convergence properties.
method Constructing families of maxfaces and minfaces with increasing cuspidal crosscaps.
result Maxfaces and minfaces converge to surfaces with folded singularities.
Geodesics near singularities either hit or wind around, with winding number dependent on singularity type.
problem Understanding geodesic behavior near singularities in Riemannian manifolds.
method Analytical study of geodesics on Riemannian manifolds near singularities.
result The winding number of geodesics around a singularity depends on the singularity type and approaches infinity as the singularity becomes cuspidal.
Study shows convergence speed for Fekete points on specific sets.
problem Understanding convergence speed for Fekete points on certain sets.
method Demonstrates (Cα,Cα′)-regularity for uniformly polynomially cuspidal sets. result Established convergence speed for Fekete points on these sets.
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.
Formula conjectured for rational cuspidal curves in projective plane.
problem Counting rational cuspidal curves in projective plane.
method Extending Kontsevich's recursion formula and using geometric input about tangency of curves at nodal points.
result Conjectural formula agrees with earlier computations and extends to rational quartics with E6 singularity.
We study surfaces of constant positive Gauss curvature in Euclidean 3-space via the harmonicity of the Gauss map. Using the loop group representation, we solve the regular and the singular geometric Cauchy problems for these surfaces, and use these solutions to compute several new examples. We give the criteria on the …
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.
We study rational cuspidal curves in Hirzebruch surfaces. We provide two obstructions for the existence of rational cuspidal curves in Hirzebruch surfaces with prescribed types of singular points. The first result comes from Heegaard--Floer theory and is a generalization of a result by Livingston and the first author. …
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.
This note gives a correction to the proof of the main result of "Harmonic representatives for cuspidal cohomology classes" by J. Dodziuk, J. McGowan and Peter Perry, an article that appeared in Serge Lang memorial volume.
The paper extends Bour's theorem to helicoidal surfaces with singularities.
problem Proving non-trivial isometric deformations for cuspidal edges under helicoidal motion.
method Generalizing Bour's theorem techniques, proving deformations for generic cuspidal edges.
result Geometric invariants are extrinsic for cuspidal edges under helicoidal motion.
Lecture notes on Heegaard Floer homologies and curves.
problem Analyzing rational cuspidal curves using Heegaard Floer homologies.
method Application of Heegaard Floer homologies to study rational cuspidal curves.
result Detailed analysis of rational cuspidal curves using Heegaard Floer homologies.
Study on singularities of Gauss maps of wave fronts with specific properties.
problem Characterizing singularities of Gauss maps of wave fronts.
method Geometric properties and boundedness of Gaussian curvatures.
result Relation between boundedness of Gaussian curvatures and types of singularities of Gauss maps.
We study the discriminant of a degree 4 extension given by a deformed bidouble cover, i.e., by equations z^2= u + a w, w^2= v + bz. We first show that the discriminant surface is a quartic which is cuspidal on a twisted cubic, i.e.,is the discriminant of the general equation of degree 3. We then take a(u,v), b(u,v) and…
The paper links set cuspidality to function regularity and flatness.
problem Linking set cuspidality to function regularity and flatness.
method Analyzes arc-smooth functions and their properties on various sets.
result Establishes a precise link between set cuspidality and function regularity.
The paper classifies symplectic invariants of specific singularities in integrable Hamiltonian systems.
problem Classifying symplectic invariants of singularities in integrable Hamiltonian systems.
method Smooth C∞ symplectic classification of Lagrangian fibrations near singularities. result Action variables form complete C∞ symplectic invariants for parabolic orbits and cuspidal tori. 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…
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 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.
We establish a twistor correspondence between a cuspidal cubic curve in a complex projective plane, and a co-calibrated homogeneous G2 structure on the seven--dimensional parameter space of such cubics. Imposing the Riemannian reality conditions leads to an explicit co-calibrated G2 structure on SU(2,1)/U(1). …
The paper studies the zeta-regularized determinant of a pseudo-Laplacian on a cuspidal end with flat unitary line bundle.
problem Analyzing the asymptotic behavior of the zeta-regularized determinant of a pseudo-Laplacian.
method Investigates the asymptotic behavior of the zeta-regularized determinant of a pseudo-Laplacian ΔL,0+μ as μ and a vary. result Determines the asymptotic behavior of the zeta-regularized determinant for various values of μ and a.