Related papers: Singularities of parallel surfaces
We consider smooth 1-parameter families of plane curves tangent to a semicubic parabola, when the curvature radius of their curves at the tangency point vanishes at the cusp point. We find the $\A$-normal form of these families, their…
When a connected component of the set of singular points of the maxface $X$ consists of only generalized cone-like singular points, we construct a sequence of maxfaces $X_n$, with an increasing number of swallowtails, converging to the…
It is shown that the geometry of parallelizable manifolds can be extended to non-parallelizable ones by extending the connection that a global frame field would define on a parallelizable manifold to a connection that a singular frame field…
We study symplectic deformation types of minimal symplectic fillings of links of quotient surface singularities. In particular, there are only finitely many symplectic deformation types for each quotient surface singularity.
We classify complete biharmonic surfaces with parallel mean curvature vector field and non-negative Gaussian curvature in complex space forms.
The node-opening technique, originally designed for constructing minimal surfaces, is adapted to construct a rich variety of new maxfaces of high genus that are embedded outside a compact set and have arbitrarily many catenoid or planar…
The goal of this work is to study the smoothings of singular coaxial intersections of ellipsoids (where coaxial includes concentric) with generic singularities, with special attention to the 3-dimensional case.
To study a deformation of a singularity taking into consideration their differential geometric properties, a form representing the deformation using only diffeomorphisms on the source space and isometries of the target space plays a crucial…
Meridian surfaces in the Euclidean 4-space are two-dimensional surfaces which are one-parameter systems of meridians of a standard rotational hypersurface. On the base of our invariant theory of surfaces we study meridian surfaces with…
Let $C$ be an irreducible projective plane curve in the complex projective space ${\mathbb{P}}^2$. The classification of such curves, up to the action of the automorphism group $PGL(3,{\mathbb{C}})$ on ${\mathbb{P}}^2$, is a very difficult…
We investigate slicings of combinatorial manifolds as properly embedded co-dimension 1 submanifolds. A focus is given to dimension 3 where slicings are normal surfaces. In the case of 2-neighborly 3-manifolds and quadrangulated slicings, a…
Near a singular point of a surface or a curve, geometric invariants diverge in general, and the orders of diverge, in particular the boundedness about these invariants represent geometry of the surface and the curve. In this paper, we study…
We study phase portraits and singular points of vector fields of a special type, that is, vector fields whose components are fractions with a common denominator vanishing on a smooth regular hypersurface in the phase space. We assume also…
We develop recursive formulas for the horizontal and vertical monodromies of a quasi-ordinary surface. These are monodromies associated to the Milnor fiber of a slice transverse to a component of the singular locus. In the course of working…
We give bounds on the gap functions of the singularities of a cuspidal plane curve of arbitrary genus, generalising recent work of Borodzik and Livingston. We apply these inequalities to unicuspidal curves whose singularity has one Puiseux…
The topological structure of the lines of principal curvature, the umbilic and partially umbilic singularities of all tridimensional ellipsoids of ${\mathbb R}^4$ is described.
In this paper the singular hypersurfaces in $\mathbb{C}\mathrm{P}^4$ of degree $d$ with an isolated singularity are studied. If the singularity is of type $A_{2k+1}$, under the condition $d<(k+5)/2$, a classification of such hypersurfaces…
Polyhedral surfaces are fundamental objects in architectural geometry and industrial design. Whereas closeness of a given mesh to a smooth reference surface and its suitability for numerical simulations were already studied extensively, the…
The main purpose of this paper is twofold. We first want to analyze in details the meaningful geometric aspect of the method introduced in the previous paper [12], concerning regularity of families of irreducible, nodal "curves" on a…
We study surface knots in 4-space by using generic planar projections. These projections have fold points and cusps as their singularities and the image of the singular point set divides the plane into several regions. The width (or the…