arXiv research

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

Classifies normal stable Horikawa surfaces with smoothable singularities.

In this paper we show that if the minimal good resolution graph of a normal surface singularity contains at least two nodes (i.e. vertex with valency at least 3) then the singularity does not admit a smoothing with Milnor fiber having rational homology equal to the rational homology of the 4-disk $D^4$ (called a ration…

Flat semigroups can represent normal weighted homogeneous surface singularities.

This paper studies the normalized Ricci flow on surfaces with conical singularities. It's proved that the normalized Ricci flow has a solution for a short time for initial metrics with conical singularities. Moreover, the solution makes good geometric sense. For some simple surfaces of this kind, for example, the tear …

It has long been known that every quasi-homogeneous normal complex surface singularity with Q-homology sphere link has universal abelian cover a Brieskorn complete intersection singularity. We describe a broad generalization: First, one has a class of complete intersection normal complex surface singularities called "s…

The paper relates curvature loci of different manifold types through projections and normal sections.

Study geometric properties of surfaces with specific formulae.

It is constructed a normal form for a class of real-smooth surfaces M\subset\mathbb{C}^{2} defined near a degenerate CR singularity.

We prove that if a contact 3-manifold admits an open book decomposition of genus 0, a certain intersection pattern cannot appear in the homology of any of its minimal symplectic fillings, and moreover, fillings cannot contain symplectic surfaces of positive genus. Applying these obstructions to canonical contact struct…

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 …

In this paper we characterize logarithmic surfaces which admit Kähler-Einstein metrics with negative scalar curvature and small edge singularities along a normal crossing divisor.

The paper studies circular evolutes and involutes of framed curves in Euclidean space.

We interpret a normal surface in a (singular) three-manifold in terms of the homology of a chain complex. This allows us to study the relation between normal surfaces and their quadrilateral co-ordinates. Specifically, we give a proof of an (unpublished) observation independently given by Casson and Rubinstein saying t…

We describe a natural decomposition of a normal complex surface singularity $(X,0)$ into its "thick" and "thin" parts. The former is essentially metrically conical, while the latter shrinks rapidly in thickness as it approaches the origin. The thin part is empty if and only if the singularity is metrically conical; the…

We introduce the notion of a ribbon-clasp surface-link, which is a generalization of a ribbon surface-link. We generalize the notion of a normal form on embedded surface-links to the case of immersed surface-links and prove that any (immersed) surface-link can be described in a normal form. It is known that an embedded…

New curvature defined for corank 1 singular surfaces in 3D.

In this paper, we study the (normalized) Ricci flow on surfaces with conical singularities. Long time existence is proved for cone angle smaller than $2π$. In this case, convergence results are obtained if the Euler number is nonpositive.

We study topological structures of the sets $(0,1/2)^3 \cap Ω$ and $(0,1/2)^3 \setminus Ω$, where~$Ω$ is one special algebraic surface defined by a symmetric polynomial in variables $a_1,a_2,a_3$ of degree~$12$. These problems arise in studying of general properties of degenerate singular points of dynamical systems ob…

Study on helicoidal singular minimal surfaces with specific properties.

Extends Kummer's theory to singular surfaces for line congruences.

We study the geometry of surfaces in $\mathbb{R}^{4}$ with corank $1$ singularities. For such surfaces the singularities are isolated and at each point we define the curvature parabola in the normal space. This curve codifies all the second order information of the surface. Also, using this curve we define asymptotic a…

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.

A generalised Thurston-Bennequin invariant for a Q-singularity of a real algebraic variety is defined as a linking form on the homologies of the real link of the singularity. The main goal of this paper is to present a method to calculate the linking form in terms of the very good resolution graph of a real normal unib…

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

The study simplifies complex functions on surfaces using a special transformation.

This is a survey article on recognition problem of frontal singularities. We specify geometrically several frontal singularities and then we solve the recognition problem of such singularities, giving explicit normal forms. We combine the recognition results by K. Saji and several arguments on openings, which was perfo…

We show that every negative definite configuration of symplectic surfaces in a symplectic 4--manifold has a strongly symplectically convex neighborhood. We use this to show that, if a negative definite configuration satisfies an additional negativity condition at each surface in the configuration, and if the complex si…

We study fundamental groups of projective varieties with normal crossing singularities and of germs of complex singularities. We prove that for every finitely-presented group G there is a complex projective surface S with simple normal crossing singularities only, so that the fundamental group of S is isomorphic to G. …

Study Stein and Milnor fillings of links from surface singularities.

Study of light function singularities on surfaces.

The paper calculates the dimension of the image of the Abel map for normal surface singularities.

While the topological types of {normal} surface singularities with homology sphere link have been classified, forming a rich class, until recently little was known about the possible analytic structures. We proved in [Geom. Topol. 9(2005) 699-755] that many of them can be realized as complete intersection singularities…

Quantum invariants of 3-manifolds linked to splice diagrams.

The normal Gauss map of a minimal surface in the model space Sol of solvegeometry is a harmonic map with respect to a certain singular Riemannian metric on the extended complex plane.

Analytic lattice cohomology defined for isolated singularities, linking to Heegaard Floer cohomology.

For any negative definite plumbed 3-manifold M we construct from its plumbed graph a graded Z[U]-module. This, for rational homology spheres, conjecturally equals the Heegaard-Floer homology of Ozsvath and Szabo, but it has even more structure. If M is a complex singularity link then the normalized Euler-characteristic…

Study delta invariant of minimal generic curves on rational surfaces.

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…

Classifies surfaces with great and small circles through each point.

We study the flat geometry of the least degenerate singularity of a singular surface in $\mathbb R^4$, the $I_{1}$ singularity parametrised by $(x,y)\mapsto(x,xy,y^{2},y^{3})$. This singularity appears generically when projecting a regular surface in $\mathbb R^5$ orthogonally to $\mathbb R^4$ along a tangent direction…

The study connects surface geometry in 5D to 4D projections and umbilic curvatures.

We show that the "geometric models of matter" approach proposed by the first author can be used to construct models of anyon quasiparticles with fractional quantum numbers, using 4-dimensional edge-cone orbifold geometries with orbifold singularities along embedded 2-dimensional surfaces. The anyon states arise through…

Classifies degenerations of complex projective plane with rational singularities.

We define cuspidal curvature $κ_c$ (resp. normalized cuspidal curvature $μ_c$) 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 $κ_Π$ called the product curva…

We survey the use of continued fraction expansions in the algebraical and topological study of complex analytic singularities. We also prove new results, firstly concerning a geometric duality with respect to a lattice between plane supplementary cones and secondly concerning the existence of a canonical plumbing struc…

We prove that the link of a complex normal surface singularity is an L--space if and only if the singularity is rational. This via a recent result of Hanselman, J. Rasmussen, S. D. Rasmussen and Watson (proving the conjecture of Boyer, Gordon and Watson), shows that a singularity link is not rational if and only if its…

In this paper, we settle in the affirmative the Jakobson-Levitin-Nadirashvili-Nigam-Polterovich conjecture, stating that a certain singular metric on the Bolza surface, with area normalized, should maximize the first eigenvalue of the Laplacian.