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We discuss the bifurcation structure of homoclinic orbits in bimodal one dimensional maps. The universal structure of these bifurcations with singular bifurcation points and the web of bifurcation lines through the parameter space are…
We investigate vertices for plane curves with singular points. As plane curves with singular points, we consider Legendre curves (respectively, Legendre immersions) in the unit tangent bundle over the Euclidean plane and frontals…
A class of surfaces-graphs in a Riemannian 3-space with a prescribed projection of one field of principal directions onto a surface $\Pi$ is considered. A problem of determination of such surfaces when both principal curvatures are given…
Mark all vertices on a curve evolving under a family of curves obtained by intersecting a smooth surface M with the 1-parameter family of planes parallel to the tangent plane to M at a point p. Those vertices trace out a set, called the…
We study the groups of automorphisms of rational algebraic surfaces that admit a relatively minimal pencil of curves of arithmetic genus one over an algebraically closed field of arbitrary characteristic. In particular, we classify such…
It is well-known that there exists a bijection between the set of lines of the projective 3-dimensional space $P^3$ and all real points of the so-called Pl\"ucker quadric $\Psi$. Moreover one can identify each point of the Pl\"ucker…
This paper is devoted to characterizing complex projective structures defined on Riemann surface orbifolds and giving rise to injective developing maps defined on the monodromy covering of the surface (orbifold) in question. The relevance…
We study bifurcation for the constant scalar curvature equation along a one-parameter family of Riemannian metrics on the total space of a harmonic Riemannian submersion. We provide an existence theorem for bifurcation points and a…
The Eckardt hypersurface in $\mathbb{P}^{19}$ parameterizes smooth cubic surfaces with an Eckardt point, which is a point common to three of the $27$ lines on a smooth cubic surface. We describe the cubic surfaces lying on the singular…
In this paper, we establish a geometric correspondence between constant curvature one metrics with two conical singularities on $S^{2}$ and isometric immersions into Euclidean 3-space $\mathbb{E}^{3}$. Specifically, we explicitly construct…
We introduce a simple combinatorial way, which we call a rectangular diagram of a surface, to represent a surface in the three-sphere. It has a particularly nice relation to the standard contact structure on $\mathbb S^3$ and to rectangular…
Motivated by the wild behavior of isolated essential singularities in complex analysis, we study singular complex analytic vector fields $X$ on arbitrary Riemann surfaces $M$. By vector field singularities we understand zeros, poles,…
We apply the invariant theory of surfaces in the four-dimensional Euclidean space to the class of general rotational surfaces with meridians lying in two-dimensional planes. We find all minimal super-conformal surfaces of this class.
Transversal structures (also known as regular edge labelings) are combinatorial structures defined over 4-connected plane triangulations with quadrangular outer-face. They have been intensively studied and used for many applications…
We present a new formulation of some basic differential geometric notions on a smooth manifold M, in the setting of nonstandard analysis. In place of classical vector fields, for which one needs to construct the tangent bundle of M, we…
We investigate the minimal and isoperimetric surface problems in a large class of sub-Riemannian manifolds, the so-called Vertically Rigid spaces. We construct an adapted connection for such spaces and, using the variational tools of…
This is an investigation into a classification of embeddings of a surface in Euclidean $3$-space. Specifically, we consider $\mathbb{R}^3$ as having the product structure $\mathbb{R}^2 \times \mathbb{R}$ and let $\pi:\mathbb{R}^2 \times…
We consider a steady axisymmetric solution of the Euler equations for a fluid (incompressible and with zero vorticity) with a free surface, acted on only by gravity. We analyze stagnation points as well as points on the axis of symmetry. At…
We study a one-parameter family of time-reversible Hamiltonian vector fields in $\mathbb{R}^4$, which has received great attention in the literature. On the one hand, it is due to the role it plays in the context of certain applications in…
Topological transitions in bubbling half-BPS Type IIB geometries with SO(4) x SO(4) symmetry can be decomposed into a sequence of n elementary transitions. The half-BPS solution that describes the elementary transition is seeded by a phase…