arXiv research

Fold singular points play important roles in the theory of maximal surfaces. For example, if a maximal surface admits fold singular points, it can be extended to a timelike minimal surface analytically. Moreover, there is a duality between conelike singular points and folds. In this paper, we investigate fold singular …

Rectifies flat singular points of area-minimizing currents with singularity degree > 1.

Study geometric properties of surfaces with specific formulae.

Study behavior of curvatures near singular points of frontals.

Study on singular points of translation surfaces under linearly dependent conditions.

The study examines singularities and geometric properties of surfaces derived from frontals with specific singular points.

The paper examines geometric invariants near a specific type of singular point.

The paper derives Gauss-Bonnet formulas for mappings between surfaces with boundary.

Study on flat singular points of area-minimizing currents, defining a singularity degree.

The paper proves ideal triangulations and disk unfolding for singular flat surfaces.

Proves conditions for positive scalar curvature on certain manifolds with conical singularities.

Study singularities of CMC 1 surfaces in de Sitter space.

Singularities of even smooth functions are studied. A classification of singular points which appear in typical parametric families of even functions with at most five parameters is given. Bifurcations of singular points near a caustic value of the parameter are also studied. A determinant for singularity types and con…

Symplectic classification for a specific type of singularity in integrable systems.

Study oscillatory integrals with degenerate singular points in multivariable phase functions.

Study helicoidal surfaces with singular points using frontals.

Research describes all possible gradient vector fields on a sphere with up to ten singular points.

Level set flow's singularities are type I under 2-convexity, leading to specific curvature blow-up rates.

Mean curvature flow shows a surface fattening at its first singular point.

Study on singularities of Lagrangian immersions with applications in Floer theory.

Study shows asymptotic behavior of metric near singular points of a Monge-Ampère equation.

Study of flows with a single singular point on a 2D disk.

Milnor fibrations were extended by Mutsuo Oka for certain mixed polynomial. In this paper, we study singular points of differentiable maps into the 2-dimensional torus, called Milnor fibration product maps, obtained by several Milnor fibrations for mixed polynomial. We give a characterization of singular points of such…

Swallowtails form on maxfaces converging to cone-like singularities.

In this article we compute the best Sobolev constants for various Hardy-Sobolev inequalities with sharp Hardy term. This is carried out in three different environments: interior point singularity in Euclidean space, interior point singularity in hyperbolic space and boundary point singularity in Euclidean domains.

We study singularities of Gauss maps of fronts and give characterizations of types of singularities of Gauss maps by geometric properties of fronts which are related to behavior of bounded principal curvatures. Moreover, we investigate relation between a kind of boundedness of Gaussian curvatures near cuspidal edges an…

The study bounds Hausdorff measure of flat singular points in area-minimizing currents.

The Kähler-Ricci flow near conical singularities is described with a $C/t$ curvature bound.

First we review the definition of a negative point mass singularity. Then we examine the gravitational lensing effects of these singularities in isolation and with shear and convergence from continuous matter. We review the Inverse Mean Curvature Flow and use this flow to prove some new results about the mass of a sing…

We prove that the asymptotic completion of a developable Möbius strip in Euclidean three-space must have at least one singular point other than cuspidal edge singularities. Moreover, if the strip contains a closed geodesic, then the number of such singular points is at least three. These lower bounds are both sharp.

In this paper, we discuss the recognition problem for A_k-type singularities on wave fronts. We give computable and simple criteria of these singularities, which will play a fundamental role in generalizing the authors' previous work "the geometry of fronts" for surfaces. The crucial point to prove our criteria for A_k…

Maps with boundary definite fold points restrict manifold structure.

Local singularity analysis for Ricci flows with applications to bounded scalar curvature.

Rectifies flat singular points for area-minimizing currents.

Milnor fibrations have been studied since 1960's. In this paper, we study singular points of differentiable maps, called Milnor fibration product maps, obtained by several Milnor fibrations. We give a characterization of singular points of such product maps, and for the case of certain weighted homogeneous polynomials,…

Theorems on the existence of vector fields with given sets of Indexes of isolated Singular points are proved for the cases of closed manifolds, pairs of manifolds, manifolds with boundary, and gradient fields. It is proved that, on a two-dimensional manifold, an index of an isolated Singular point of the gradient field…

The paper studies singularities of pedal curves of hyperbolic frontals.

This paper explores how local behavior of meromorphic connections on the projective line determines the global connection.

At each point in an immersed surface in $\mathbb R^4$ there is a curvature ellipse in the normal plane which codifies all the local second order geometry of the surface. More recently, at the singular point of a corank 1 singular surface in $\mathbb R^3$, a curvature parabola in the normal plane which codifies all the …

Ricci flows with bounded scalar curvature cannot develop Type I singular points.

We study the stability of singular points for smooth Poisson structures as well as general Lie algebroids. We give sufficient conditions for stability lying on the first (not necessarily linear) approximation of the given Poisson structure or Lie algebroid at a singular point. The main tools used here are the classical…

The paper classifies vertices in planar polygons formed by convex domains.

This paper continues the previous studies in two papers of Huang-Yin [HY3-4] on the flattening problem of a CR singular point of real codimension two sitting in a submanifold in ${\mathbb C}^{n+1}$ with $n+1\ge 3$, whose CR points are non-minimal. Partially based on the geometric approach initiated in [HY3] and a forma…

In dimension 7, we establish a Fredholm theory for a Dirac-type operator associated to a connection with point singularities. There are two applications. $1$. over a closed 7-manifold, under some natural conditions, a $G_{2}-$instanton and its point singularities can still be "seen" when the $G_{2}-$structure is proper…

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. …

We study and completely describe pairs of compatible Poisson structures near singular points of the recursion operator satisfying natural non-degeneracy condition.

Algorithm classifies saddle-focus singularities in Hamiltonian systems.

Solves Deligne-Simpson problem for special connections on Gm.