Legendrian graphs generated by Tangential Families

Simple tangential families and perestroikas of their envelopes

Tangential families are 1-parameter families of rays emanating tangentially from smooth curves. We classify tangential family germs up to Left-Right equivalence: we prove that there are two infinite series and four sporadic simple singularities of tangential family germs (in addition to two stable singularities). We give their normal forms and miniversal tangential deformations (i.e., deformations among tangential families), and we describe the corresponding envelope perestroikas of small codimension. We also discuss envelope singularities of non simple tangential families.

Stable tangential families and singularities of their envelopes

We study tangential families, i.e. systems of rays emanating tangentially from given curves. We classify, up to Left-Right equivalence, stable singularities of tangential family germs (under deformations among tangential families) and we study their envelopes. We discuss applications of our results to the case of tangent geodesics of a curve.

不能参数化地成为Legendrian的纽结族

Families of knots that cannot be made Legendrian parametrically

本文证明对于每个n≥3，任何纽结类型的Legendrian代表及其形式代表在π_n层面上的群同态均非满射，并指出π_2层面的满射性依赖纽结类型，从而揭示了参数族在更高同伦层面的刚性。

Generalization of the Chekanov theorem: diameters of immersed manifolds and wave fronts

The Chekanov theorem generalizes the classic Lyusternik-Shnirel'man and Morse theorems concerning critical points of a smooth function on a closed manifold. A Legendrian submanifold Λof space of 1-jets of the functions on a manifold M defines a multi-valued function whose graph is the projection of Λin J^0 M = M x R. The Chekanov theorem asserts that if Λis homotopic to the 1-jet of a smooth function in the class of embedded Legendrian manifolds, then such a graph of a multi-valued function must have a lot of points (their number is determined by the topology of M) at which the tangent plane t…

Moduli of Legendrian foliations and quadratic differentials in the Heisenberg group

The aim of the paper is to prove the following result concerning moduli of curve families in the Heisenberg group. Let $Ω$ be a domain in the Heisenberg group foliated by a family $Γ$ of legendrian curves. Assume that there is a quadratic differential $q$ on $Ω$ in the kernel of an operator defined in \cite{Tim2} and every curve in $Γ$ is a horizontal trajectory for $q$. Let $l_Γ: Ω\rightarrow ]0,+\infty[$ be the function that associates to a point $p\in Ω$, the $q$-length of the leaf containing $p$. Then, the modulus of $Γ$ is \[ M_4 (Γ) = \int_Ω\frac{|q|^2}{(l_Γ) ^4} \mathrm{d} L^3.\]

Vertices and inflexions of plane sections of surfaces in R^3

We investigate the behaviour of vertices and inflexions on 1-parameter families of curves on smooth surfaces in the 3-space, which include a singular member. In particular, we discuss the context where the curves evolve as sections of a smooth surface by parallel planes. More precisely we will trace the patterns of inflexions and vertices (maxima and minima of curvature) on the sections of a surface as the section passes through a tangential point. We also keep track on the evolution of the curvature of the curves at vertices and control its limit when the vertices collapse at singular points.…