Legendrian Submanifolds Research Articles

On the topology of loops of contactomorphisms and Legendrians in non-orderable manifolds

We study the global topology of the space L of loops of contactomorphisms of a non-orderable closed contact manifold (M2n+1,ξ). We filter L by a quantitative measure of the “positivity” of the loops and describe the topology of L in terms of the subspaces of the filtration. In particular, we show that the homotopy groups of L are subgroups of the homotopy groups of the subspace of positive loops L+. We obtain analogous results for the space of loops of Legendrian submanifolds in (M2n+1,ξ).

On thermodynamic processes, state equations and critical phenomena for homogeneous mixtures

In this paper, we study the thermodynamics of homogeneous mixtures in equilibrium. From the perspective of thermodynamics, substances are understood as Legendre submanifolds, which are equipped with a Riemannian structure in addition. We refer to these as Legendre-Riemannian manifolds. This Legendre structure reflects the law of conservation of energy, while the Riemannian structure corresponds to the second central moment of measurement of extensive quantities, indicating that we only consider stable states. Thermodynamic processes, such as chemical reactions, correspond to contact vector fields that preserve the law of energy conservation, or are contact. The presence of a Riemannian structure distinguishes between three classes of processes: positive, which increase the metric; neutral, which preserve the metric; and negative, which decrease the metric. We provide a detailed description of the processes and suggest a method for finding state equations for a homogeneous mixture in mechanical or chemical equilibrium.

Generalized Wintgen ideal Legendrian submanifolds of generalized Sasakian-space forms

Generalized Wintgen ideal Legendrian submanifolds of generalized Sasakian-space forms

Existence of generating families on Lagrangian cobordisms

For an embedded exact Lagrangian cobordism between Legendrian submanifolds in the 1-jet bundle, we prove that a generating family linear at infinity on the Legendrian at the negative end extends to a generating family linear at infinity on the Lagrangian cobordism after stabilization if and only if the formal obstructions vanish. In particular, a Lagrangian filling with trivial stable Lagrangian Gauss map admits a generating family linear at infinity.

On conformally flat minimal Legendrian submanifolds in the unit sphere

This paper is concerned with the study on an open problem of classifying conformally flat minimal Legendrian submanifolds in the $(2n+1)$ -dimensional unit sphere $\mathbb {S}^{2n+1}$ admitting a Sasakian structure $(\varphi,\,\xi,\,\eta,\,g)$ for $n\ge 3$ , motivated by the classification of minimal Legendrian submanifolds with constant sectional curvature. First of all, we completely classify such Legendrian submanifolds by assuming that the tensor $K:=-\varphi h$ is semi-parallel, which is introduced as a natural extension of $C$ -parallel second fundamental form $h$ . Secondly, such submanifolds have also been determined under the condition that the Ricci tensor is semi-parallel, generalizing the Einstein condition. Finally, as direct consequences, new characterizations of the Calabi torus are presented.

Chen inequalities on warped product Legendrian submanifolds in Kenmotsu space forms and applications

In the current work, we study the geometry and topology of warped product Legendrian submanifolds in Kenmotsu space forms F2n+1(ϵ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{F}^{2n+1}(\\epsilon )$\\end{document} and derive the first Chen inequality, including extrinsic invariants such as the mean curvature and the length of the warping functions. Additionally, sectional curvature and the δ-invariant are intrinsic invariants related to this inequality. An integral bound is also given in terms of the gradient Ricci curvature for the Bochner operator formula of compact warped product submanifolds. Our primary technique is applying geometry to number structures and solving problems such as problems with Dirichlet eigenvalues.

Brane structures in microlocal sheaf theory

AbstractLet be an exact Lagrangian submanifold of a cotangent bundle , asymptotic to a Legendrian submanifold . We study a locally constant sheaf of ‐categories on , called the sheaf of brane structures or . Its fiber is the ‐category of spectra, and we construct a Hamiltonian invariant, fully faithful functor from to the ‐category of sheaves of spectra on with singular support in .

On Newton polytopes of Lagrangian augmentations

AbstractThis note explores the use of Newton polytopes in the study of Lagrangian fillings of Legendrian submanifolds. In particular, we show that Newton polytopes associated to augmented values of Reeb chords can distinguish infinitely many distinct Lagrangian fillings, both for Legendrian links and higher dimensional Legendrian spheres. The computations we perform work in finite characteristic, which significantly simplifies arguments and also allows us to show that there exist Legendrian links with infinitely many nonorientable exact Lagrangian fillings.

An Invariant of Riemannian Type for Legendrian Warped Product Submanifolds of Sasakian Space Forms

In the present paper, we investigate the geometry and topology of warped product Legendrian submanifolds in Sasakian space forms D2n+1(ϵ) and obtain the first Chen inequality that involves extrinsic invariants like the mean curvature and the length of the warping functions. This inequality also involves intrinsic invariants (δ-invariant and sectional curvature). In addition, an integral bound is provided for the Bochner operator formula of compact warped product submanifolds in terms of the gradient Ricci curvature. Some new results on mean curvature vanishing are presented as a partial solution to the well-known problem given by S.S. Chern.

Implicit contact dynamics and Hamilton-Jacobi theory

In this paper, we introduce implicit Hamiltonian dynamics in the framework of contact geometry in two different ways: first, we introduce classical implicit Hamiltonian dynamics on a contact manifold, followed by evolution Hamiltonian dynamics. In the first case, implicit contact Hamiltonian dynamics is defined as a Legendrian submanifold of a tangent contact space, whilst the implicit evolution dynamic is understood as a Lagrangian submanifold of a certain symplectic space embedded into the tangent contact space. To conclude, we propose a geometric Hamilton-Jacobi theory for both of these formulations.

On torsion in linearized Legendrian contact homology

In this short note, we discuss certain examples of Legendrian submanifolds, whose linearized Legendrian contact (co)homology groups over integers have non-vanishing algebraic torsion. More precisely, for a given arbitrary finitely generated abelian group [Formula: see text] and a positive integer [Formula: see text], [Formula: see text], we construct examples of Legendrian submanifolds of the standard contact vector space [Formula: see text], whose [Formula: see text]th linearized Legendrian contact (co)homology over [Formula: see text] computed with respect to a certain augmentation is isomorphic to [Formula: see text].

Contact topology and non-equilibrium thermodynamics

We describe a method, based on contact topology, of showing the existence of semi-infinite trajectories of contact Hamiltonian flows which start on one Legendrian submanifold and asymptotically converge to another Legendrian submanifold. We discuss a mathematical model of non-equilibrium thermodynamics where such trajectories play a role of relaxation processes, and illustrate our results in the case of the Glauber dynamics for the mean field Ising model.

Minimal Legendrian submanifolds in Sasakian space forms with C-parallel second fundamental form

It is known that Cheng-He-Hu [3] gave a complete classification of the n-dimensional C-totally real (or equivalently, Legendrian) minimal submanifolds in the (2n+1)-dimensional Sasakian space form N2n+1(c) with constant sectional curvature. According to this classification and the proof therein, such minimal Legendrian submanifolds are all of C-parallel second fundamental form. The purpose of this paper is to extend the above result partly, and as the main results, we classify all the minimal Legendrian submanifolds in N2n+1(c) with C-parallel second fundamental form in cases n=2,3,4.

Optimal Inequalities for Submanifolds in Trans-Sasakian Manifolds Endowed with a Semi-Symmetric Metric Connection

In this paper, we improve the Chen first inequality for special contact slant submanifolds and Legendrian submanifolds, respectively, in (α,β) trans-Sasakian generalized Sasakian space forms endowed with a semi-symmetric metric connection.

Legendrian submanifolds from Bohr–Sommerfeld covers of monotone Lagrangian tori

Legendrian submanifolds from Bohr–Sommerfeld covers of monotone Lagrangian tori

On the Roter type of generalised Wintgen ideal Legendrian submanifolds

I. Mihai obtained an inequality relating intrinsic normalised scalar curvature and extrinsic squared mean curvature and normalised normal curvature of Legendrian submanifolds Mn in Sasakian space forms eM2n+1(c). In this paper, for the class of generalised Wintgen ideal Legendrian submanifolds Mn of Sasakian space form eM2n+1(c), we study relationship between some properties concerning their Deszcz symmetry and their Roter type.

Rigidity of closed CSL submanifolds in the unit sphere

We are concerned with the rigidity of contact stationary Legendrian (CSL) submanifolds, critical points of the volume functional of Legendrian submanifolds in a Sasakian manifold, whose Euler–Lagrange equation is a third-order elliptic PDE. We obtain several optimal rigidity theorems for closed CSL submanifolds in the unit sphere by utilizing the maximum principle together with Simons’ identity. In particular, we proved that a closed CSL submanifold M^{n}\subset\mathbb{S}^{2n+1} is a totally geodesic sphere or a Calabi 2 -torus if |{\mathbf{B}}|^2\leq\frac{4(n-1)}{n}+\frac{3n-2}{n^2}|{\mathbf{H}}|^2 , where \mathbf{B} and \mathbf{H} are the second fundamental form and the mean curvature vector, respectively. Moreover, an example shows that this assumption is optimal.

An optimal pinching theorem of minimal Legendrian submanifolds in the unit sphere

An optimal pinching theorem of minimal Legendrian submanifolds in the unit sphere

A note on the infinite number of exact Lagrangian fillings for spherical spuns

In this short note we discuss high-dimensional examples of Legendrian submanifolds of the standard contact Euclidean space with an infinite number of exact Lagrangian fillings up to Hamiltonian isotopy. They are obtained from the examples of Casals and Ng by applying to them the spherical spinning construction.

The fundamental theorem of Legendrian submanifolds in the Heisenberg group

We study uniqueness and existence questions for Legendrian submanifolds in the Heisenberg group via Cartan's method of moving frames and the theory of Lie groups. Moreover, we generalize this result to any dimensional isotropic submanifolds in the Heisenberg group.