Contact Submanifolds Research Articles

The Legendrian Whitney trick

In this article, we prove a Legendrian Whitney trick which allows for the removal of intersections between codimension-two contact submanifolds and Legendrian submanifolds, assuming such a smooth cancellation is possible. This technique is applied to show the existence h-principle for codimension-two contact embeddings with a prescribed contact structure.

Non-simplicity of Isocontact Embeddings in All Higher Dimensions

In this article we show that in any dimension there exist infinitely many pairs of formally contact isotopic isocontact embeddings into the standard contact sphere which are not contact isotopic. This is the first example of rigidity for contact submanifolds in higher dimensions. The contact embeddings are constructed via contact push-offs of higher-dimensional Legendrian submanifolds, a construction that generalizes the union of the positive and negative transverse push-offs of a Legendrian knot to higher dimensions.

Tight neighborhoods of contact submanifolds

Tight neighborhoods of contact submanifolds

Floer cohomology, multiplicity and the log canonical threshold

Let f be a polynomial over the complex numbers with an isolated singularity at 0. We show that the multiplicity and the log canonical threshold of f at 0 are invariants of the link of f viewed as a contact submanifold of the sphere. This is done by first constructing a spectral sequence converging to the fixed point Floer cohomology of any iterate of the Milnor monodromy map whose E^1 page is explicitly described in terms of a log resolution of f. This spectral sequence is a generalization of a formula by A'Campo. By looking at this spectral sequence, we get a purely Floer theoretic description of the multiplicity and log canonical threshold of f.

Homotopy classification of contact foliations on open contact manifolds

We give a homotopy classification of foliations on open contact manifolds whose leaves are contact submanifolds of the ambient space. The result is an extension of Haefliger’s classification of foliations on open manifold in the contact setting. While proving the main theorem, we also prove a result on equidimensional isocontact immersions on open contact manifolds.

On the Contact Geometry and the Poisson Geometry of the Ideal Gas.

We elaborate on existing notions of contact geometry and Poisson geometry as applied to the classical ideal gas. Specifically, we observe that it is possible to describe its dynamics using a 3-dimensional contact submanifold of the standard 5-dimensional contact manifold used in the literature. This reflects the fact that the internal energy of the ideal gas depends exclusively on its temperature. We also present a Poisson algebra of thermodynamic operators for a quantum-like description of the classical ideal gas. The central element of this Poisson algebra is proportional to Boltzmann’s constant. A Hilbert space of states is identified and a system of wave equations governing the wavefunction is found. Expectation values for the operators representing pressure, volume and temperature are found to satisfy the classical equations of state.

An obstruction for codimension two contact embeddings in the odd dimensional Euclidean spaces

We prove that the first Chern class of a codimension two closed contact submanifold of the odd dimensional Euclidean space is trivial. For any closed co-oriented contact 3-manifold with trivial Chern class, we prove that there is a contact structure on the 5-dimensional Euclidean space which admits a contact embedding of it.

On the Links of Simple Singularities, Simple Elliptic Singularities and Cusp Singularities

Abstract This is a survey article about the study of the links of some complex hypersurface singularities in ℂ3 . We study the links of simple singularities, simple elliptic singularities and cusp singularities, and the canonical contact structures on them. It is known that each singularity link is diffeomorphic to a compact quotient of a 3-dimensional Lie group SU (2), Nil3 or Sol3 , respectively. Moreover, the canonical contact structure is equivalent to the contact structure invariant under the action of each Lie group. We show a new proof of this fact using the moment polytope of S5 . Our proof gives a new aspect to the relation between simple elliptic singularities and cusp singularities, and visualizes how the singularity links are embedded in S5 as codimension two contact submanifolds.

On contact embeddings of contact manifolds in the odd dimensional Euclidean spaces

We prove that a closed co-oriented contact (2m + 1)-manifold (M2m + 1, ξ) can be a contact submanifold of the standard contact structure on ℝ4m + 1, if it satisfies one of the following conditions: (1) m is odd (m ≥ 3) and H1(M2m + 1; ℤ) = 0, (2) m is even (m ≥ 4) and M2m + 1 is 2-connected, (3) m = 2 and M5 is simply-connected.

The Reeb foliation arises as a family of Legendrian submanifolds at the end of a deformation of the standard [formula omitted] in [formula omitted

We realize the Reeb foliation of S3 as a family of Legendrian submanifolds of the unit S5⊂C3. Moreover, we construct a deformation of the standard contact S3 in S5, via a family of contact submanifolds, into this realization.

Some remarks on the size of tubular neighborhoods in contact topology and fillability

The well-known tubular neighborhood theorem for contact submanifolds states that a small enough neighborhood of such a submanifold N is uniquely determined by the contact structure on N, and the conformal symplectic structure of the normal bundle. In particular, if the submanifold N has trivial normal bundle then its tubular neighborhood will be contactomorphic to a neighborhood of Nx{0} in the model space NxR^{2k}. In this article we make the observation that if (N,\xi_N) is a 3-dimensional overtwisted submanifold with trivial normal bundle in (M,\xi), and if its model neighborhood is sufficiently large, then (M,\xi) does not admit an exact symplectic filling.

Lefschetz type pencils on contact manifolds

We define the concept of Lefschetz contact pencil and we show the existence of such structures on any contact manifold. The main idea of the proof is a generalization of the Donaldson arguments used in the symplectic case. We will analyze some of the applications of such existence theorem for the topology of approximately holomorphic contact submanifolds.

Sur la torsion des structures de contact tendues

Thanks to torsion, an invariant introduced by E. Giroux, we give a sufficient topological condition for a 3-manifold to carry infinitely many non-isomorphic tight contact structures. It is for example the case of every orientable Seifert fibered space over a surface of genus g⩾1. We also prove a rigidity result for the existence of some contact submanifolds in a universally tight contact 3-manifold.

On the Construction of Contact Submanifolds with Prescribed Topology

We prove the existence of contact submanifolds realizing the Poincare dual of the top Chern class of a complex vector bundle over a closed contact manifold. This result is analogue in the contact categoryto Donaldson's construction of symplectic submanifolds. The main tool in the construction is to show the existence of sequences of sections which are asymptotically holomorphic in an appropiate sense and that satisfya transversalitywith es- timates propertydirectlyin the contact category . The description of the ob- tained contact submanifolds allows us to prove an extension of the Lefschetz hyperplane theorem which completes their topological characterization.

On Contact Submanifolds in Complex Projective Spaces

We treat n-dimensional real submanifolds of complex projective spaces in the case when the maximal holomorphic tangent subspace is (n - 1)-dimensional. In particular, we study the case when the induced almost contact structure on a submanifold is contact, we establish a few characteristics of the shape operator with respect to the distinguished vector field and we give one characterization of a contact submanifold in this case