arXiv research

Legendrian Lavrentiev links are shown to be equivalent to smooth links.

Algorithm converts curves on ribbon surfaces to contact surgery diagrams.

Study on cr-invariant variational problem for Legendrian curves in 3-sphere.

Study geometric mKdV flows for Legendrian curves in a 3-sphere.

Using convex integration we give a constructive proof of the well-known fact that every continuous curve in a contact $3$-manifold can be approximated by a Legendrian curve.

Constructs Lagrangian skeleta for curve singularities.

Using convex surfaces and Kanda's classification theorem, we classify Legendrian isotopy classes of Legendrian linear curves in all tight contact structures on $T^3$. Some of the knot types considered in this article provide new examples of non transversally simple knot types.

Motivated by Legendrian curve shortening flows in $\mathbb{R}^{3}$, we study the curve shortening flow of figure-eight curves in the plane. We show that, under some symmetry and curvature conditions, a figure-eight curve will shrink to a point at the first singular time.

The study finds non-simple isotopy classes of links in 3-manifolds, including Legendrian and pseudo-Legendrian examples.

The paper introduces a new flow for Legendrian curves in Sasakian sub-Riemannian 3-manifolds.

Geometrically, Legendrian surfaces related by surgery have related skein-valued cluster spaces.

Study on focal surfaces and evolutes of framed curves in hyperbolic 3-space using Legendrian duality.

New insights into $(2,3,5)$-distributions via Legendrian curves.

The paper studies singularities of pedal curves of hyperbolic frontals.

The paper studies Legendrian mean curvature flow in η-Einstein Sasakian manifolds.

We construct a Legendrian version of Envelope theory. A tangential family is a 1-parameter family of rays emanating tangentially from a smooth plane curve. The Legendrian graph of the family is the union of the Legendrian lifts of the family curves in the projectivized cotangent bundle $PT^*\R^2$. We study the singular…

We discuss Ghys' theorem on 4 zeroes of the Schwarzian derivative and its relation with flattening points of Legendrian curves and Sturm theory.

For null curves in PSL(2,C), there exists a representation formula in terms of two meromorphic functions and their derivatives (Small's formula). In this paper, we give an elementary proof of Small's formula. Moreover, a similar formula for Legendrian curves in PSL(2,C) is given. As null curves in PSL(2,C) are related …

In this paper, we find a holomorphic Darboux chart around any immersed noncompact holomorphic Legendrian curve in a complex contact manifold $(X,ξ)$. By using such a chart, we show that every holomorphic Legendrian immersion $R\to X$ from an open Riemann surface can be approximated on relatively compact subsets by holo…

We obtain a Runge approximation theorem for holomorphic Legendrian curves and immersions in the complex projective $3$-space $\mathbb{CP}^3$, both from open and compact Riemann surfaces, and we prove that the space of Legendrian immersions from an open Riemann surface into $\mathbb{CP}^3$ is path connected. We also sho…

New moves for singular knots identified and described.

New integer-valued functions for Legendrian knots.

New surfaces found in 5D space.

In this paper we study holomorphic Legendrian curves in the standard holomorphic contact structure on $\mathbb{C}^{2n+1}$ for any $n\in\mathbb{N}$. We provide several approximation and desingularization results which enable us to prove general existence theorems, settling some of the open problems in the subject. In pa…

Consider an immersed Legendrian surface in the five dimensional complex projective space equipped with the standard homogeneous contact structure. We introduce a class of fourth order projective Legendrian deformation called \emph{$\,Ψ$-deformation}, and give a differential geometric characterization of surfaces admitt…

Study shows how lengths of geodesic arcs determine linking number of Legendrian knots.

The Gronwall conjecture states that a planar 3-web of foliations which admits more than one distinct linearizations is locally equivalent to an algebraic web. We propose an analogue of the Gronwall conjecture for the 3-web of foliations by Legendrian curves in a contact three manifold. The Legendrian Gronwall conjectur…

In this paper we prove that every open Riemann surface properly embeds in the Special Linear group $SL_2(\mathbb{C})$ as a holomorphic Legendrian curve, where $SL_2(\mathbb{C})$ is endowed with its standard contact structure. As a consequence, we derive the existence of proper, weakly complete, flat fronts in the real …

Study geometrically characterizes piecewise circular curves with decreasing curvature.

The Thurston-Bennequin invariant provides one notion of self-linking for any homologically-trivial Legendrian curve in a contact three-manifold. Here we discuss related analytic notions of self-linking for Legendrian knots in Euclidean space. Our definition is based upon a reformulation of the elementary Gauss linking …

In this paper we study a subspace of the space of Legendrian loops and we show that the injection of this space into the full loop space is an S1-equivariant homotopy equivalence. This space can be also seen as the space of zero Maslov index Legendrian loops and it shows up as a suitable space of variations in contact …

Let $ X: M \hook S^5$ be a compact Legendrian surface in pseudoconformal(CR) 5-sphere. We introduce a pseudoconformally invariant Willmore type second order functional $ \W(X)$, and study its critical points called Willmore Legendrian surfaces. The fifth order structure equations show that Willmore dual can be defined …

We study those Lagrangian surfaces in complex Euclidean space which are foliated by circles or by straight lines. The former, which we call cyclic, come in three types, each one being described by means of, respectively, a planar curve, a Legendrian curve of the 3-sphere or a Legendrian curve of the anti de Sitter 3-sp…

The paper reformulates Legendrian contact homology using string topology.

Study null surfaces of pseudo-spherical curves in anti-de Sitter space.

Develops gluing theory for contact instantons and pseudoholomorphic curves.

We construct a complete, bounded Legendrian immersion in C^3. As direct applications of it, we show the first examples of a weakly complete bounded flat front in hyperbolic 3-space, a weakly complete bounded flat front in de Sitter 3-space, and a weakly complete bounded improper affine front in R^3.

New construction shows VMRTs of unbendable curves can be Legendrian.

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 e…

Worldsheet skein D-module for Hopf link conormal uniquely determines partition functions.

We prove generic fibre of Painlevé moduli spaces are Weinstein handlebodies.

Superminimal surfaces in certain Einstein manifolds have a Calabi-Yau property.

Proper superminimal surfaces in hyperbolic 4-space can be approximated by conformal immersions.

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 vertex set of M through p. We take p to be an isolated umbilic point on M and descr…

We consider evolution equations for curves in the 3-dimensional sphere $S^3$ that are invariant under the group $SU(2,1)$ of pseudoconformal transformations, which preserves the standard contact structure on the sphere. In particular, we investigate how invariant evolutions of Legendrian and transverse curves induce we…

It is shown that, in the 1-jet space of the circle, the swapping and the flyping procedures, which produce topologically equivalent links, can produce nonequivalent legendrian links. Each component of the links considered is legendrian isotopic to the 1-jet of the 0-function, and thus cannot be distinguished by the cla…

Study on evolutes and focal surfaces of pseudo-spherical framed immersions in anti-de Sitter space.

Study proves h-principles for curves in bracket-generating distributions.