Canonical Contact Structure Research Articles

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.

The Canonical Contact Structure on the Link of a Cusp Singularity

Caubel, Nemethi, and Popescu-Pampu in [2] proved that an oriented 3-manifold admits at most one positive contact structure which can be realized as the complex tangency along the link of a complex surface singularity. They call it the Milnor fillable contact structure. Lekili and Ozbagci in [10] showed that a Milnor fillable contact structure is universally tight. In particular, by Honda's classification [5], the link of a cusp singularity is contactomorphic to the positive contact structure associated to the Anosov flow on a Sol-manifold (see [1]). We describe the contact structure on the link of a cusp singularity in two different ways without using Honda's classification theorem. One description is based on the toric method introduced in Mori [15]. The other description is based on Hirzebruch's construction of the Hilbert modular cusps. Consequently, we give certain answers to the problems in Mori [14] concerning the relation between the cusp singularities and the simple elliptic singularities, and the higher dimensional extension of the local Lutz-Mori twist.

Canonical contact structures on some singularity links

We identify the canonical contact structure on the link of a simple elliptic or cusp singularity by drawing a Legendrian handlebody diagram of one of its Stein fillings. We also show that the canonical contact structure on the link of a numerically Gorenstein surface singularity is trivial considered as a real plane bundle.

On the space of light rays of a spacetime and a reconstruction theorem by Low

A reconstruction theorem in terms of the topology and geometrical structures on the spaces of light rays and skies of a given spacetime is discussed. This result can be seen as a part of Penrose and Low’s programme intending to describe the causal structure of a spacetime M in terms of the topological and geometrical properties of the space of light rays, i.e., unparametrized time-oriented null geodesics, . In the analysis of the reconstruction problem, the structure of the space of skies, i.e., of congruences of light rays, becomes instrumental. It will be shown that the space of skies Σ of a strongly causal non-refocusing spacetime M carries a canonical differentiable structure diffeomorphic to the original manifold M. Celestial curves, this is, curves in which are everywhere tangent to skies, play a fundamental role in the analysis of the geometry of the space of light rays. It will be shown that a celestial curve is induced by a past causal curve of events iff the Legendrian isotopy defined by it is non-negative. This result extends in a nontrivial way some recent results by Chernov et al on Low’s Legendrian conjecture. Finally, it will be shown that a celestial causal map between the space of light rays of two strongly causal spaces (provided that the target space is null non-conjugate) is necessarily induced from a conformal immersion and conversely. These results make explicit the fundamental role played by the collection of skies, a collection of Legendrian spheres with respect to the canonical contact structure on , in characterizing the causal structure of spacetime.

Contact Structure on Mixed Links

A strongly nondegenerate mixed function has a Milnor open book structures on a sufficiently small sphere. We introduce the notion of a holomorphic-like mixed function, and we show that a link defined by such a mixed function has a canonical contact structure. Then we will show that this contact structure for a certain holomorphic-like mixed function is carried by the Milnor open book.

Singularity links with exotic Stein fillings

In a recent paper of Akhmedov, Etnyre, Mark and Smith, it was shown that there exist infinitely many contact Seifert fibered 3-manifolds each of which admits infinitely many exotic (homeomorphic but pairwise non-diffeomorphic) simply-connected Stein fillings. Here we extend this result to a larger set of contact Seifert fibered 3-manifolds with many singular fibers and observe that these 3-manifolds are singularity links. In addition, we prove that the contact structures induced by the Stein fillings are the canonical contact structures on these singularity links. As a consequence, we verify a prediction of Andras Nemethi by providing examples of isolated complex surface singularities whose links with their canonical contact structures admitting infinitely many exotic simply-connected Stein fillings. Moreover, for infinitely many of these contact singularity links and for each positive integer n, we also construct an infinite family of exotic Stein fillings with fixed fundamental group $\mathbb{Z} \oplus \mathbb{Z}_n$.

Hidden symmetries of the gravitational contact structure of the classical phase space of general relativistic test particle

The phase space of general relativistic test particle is defined as the 1-jet space of motions. A Lorentzian metric defines the canonical contact structure on the odd-dimensional phase space. In the paper we study infinitesimal symmetries of the gravitational contact phase structure which are not generated by spacetime infinitesimal symmetries, i.e. they are hidden symmetries. We prove that Killing multivector fields admit hidden symmetries of the gravitational contact phase structure and we give the explicit description of such hidden symmetries.

$${\mathcal{N} = 2}$$ N = 2 Supersymmetric AdS4 Solutions of M-theory

We analyse the most general N=2 supersymmetric solutions of D=11 supergravity consisting of a warped product of four-dimensional anti-de-Sitter space with a seven-dimensional Riemannian manifold Y_7. We show that the necessary and sufficient conditions for supersymmetry can be phrased in terms of a local SU(2)-structure on Y_7. Solutions with non-zero M2-brane charge also admit a canonical contact structure, in terms of which many physical quantities can be expressed, including the free energy and the scaling dimensions of operators dual to supersymmetric wrapped M5-branes. We show that a special class of solutions is singled out by imposing an additional symmetry, for which the problem reduces to solving a second order non-linear ODE. As well as recovering a known class of solutions, that includes the IR fixed point of a mass deformation of the ABJM theory, we also find new solutions which are dual to cubic deformations. In particular, we find a new supersymmetric warped AdS_4 x S^7 solution with non-trivial four-form flux.

The canonical contact structure on the space of oriented null geodesics of pseudospheres and products

Abstract Let Sk,m be the pseudosphere of signature (k,m). We show that the space ℒ0(Sk,m) of all oriented null geodesics in Sk,m is a manifold, and we describe geometrically its canonical contact distribution in terms of the space of oriented geodesics of certain totally geodesic degenerate hypersurfaces in Sk;m. Further, we find a contactomorphism with some standard contact manifold, namely, the unit tangent bundle of some pseudo-Riemannian manifold. Also, we express the null billiard operator on ℒ0(Sk,m) associated with some simple regions in Sk;m in terms of the geodesic flows of spheres. For the pseudo-Riemannian product N of two complete Riemannian manifolds, we give geometrical conditions on the factors for ℒ0(N) to be manifolds and exhibit a contactomorphism with some standard contact manifold.

Graded contact manifolds and contact Courant algebroids

We develop a systematic approach to contact and Jacobi structures on graded supermanifolds. In this framework, contact structures are interpreted as symplectic principal R×-bundles. Gradings compatible with the R×-action lead to the concept of a graded contact manifold, in particular a linear (more generally, n-linear) contact structure. Linear contact structures are proven to be exactly the canonical contact structures on first jets of line bundles. They provide linear Kirillov (or Jacobi) brackets and give rise to the concept of a Kirillov algebroid, an analog of a Lie algebroid, for which the corresponding cohomology operator is represented not by a vector field (de Rham derivative) but by a first-order differential operator. It is shown that one can view Kirillov or Jacobi brackets as homological Hamiltonians on linear contact manifolds. Contact manifolds of degree 2, as well as contact analogs of Courant algebroids are studied. We define lifting procedures that provide us with constructions of canonical examples of the structures in question.

Milnor open books of links of some rational surface singularities

We describe Milnor open books and Legendrian surgery diagrams for canon- ical contact structures of links of some rational surface si ngularities. We also describe an infinite family of Milnor fillable contact 3-manifolds so that the Milnor genus (resp. Mil- nor norm) is strictly greater than the support genus (resp. s upport norm) of the canonical contact structure, for each member of this family.

The free energy of $ \mathcal{N} = 2 $ supersymmetric AdS4 solutions of M-theory

We show that general $ \mathcal{N} = 2 $ supersymmetric AdS4 solutions of M-theory with non-zero M2-brane charge admit a canonical contact structure. The free energy of the dual superconformal field theory on S 3 and the scaling dimensions of operators dual to supersymmetric wrapped M5-branes are expressed via AdS/CFT in terms of contact volumes. In particular, this leads to topological and localization formulae for the coefficient of N 3/2 in the free energy of such solutions.

Partial immersions and partially free maps

In a recent paper DʼAmbra et al. (2011) [2] we studied basic properties of partial immersions and partially free maps, a generalization of free maps introduced first by Gromov (1970) in [4]. In this short note we show how to build partially free maps out of partial immersions and use this fact to prove that the partially free maps in critical dimension introduced in Theorems 1.1–1.3 of DʼAmbra et al. (2011) [2] for three important types of distributions can actually be built out of partial immersions. Finally, we show that the canonical contact structure on R 2 n + 1 admits partial immersions in critical dimension for every n.

Invariants of open books of links of surface singularities

If M is the link of a complex normal surface singularity, then it carries a canonical contact structure ξcan, which can be identified from the topology of the 3-manifold M. We assume that M is a rational homology sphere. We compute the support genus, the binding number and the norm associated with the open books which support ζcan, provided that we restrict ourselves to the case of (analytic) Milnor open books. In order to do this, we determine monotonity properties of the genus and the Milnor number of all Milnor fibrations in terms of the Lipman cone.We generalize results of [3] valid for links of rational surface singularities, and we answer some questions of Etnyre and Ozbagci [7, section 8] regarding the above invariants.

LEGENDRIAN HELIX AND CABLE LINKS

Traynor ([11]) has described an example of a two-component Legendrian "circular helix link" Λ0 ⊔ Λ1 in the 1-jet space J1(S1) of the circle (with its canonical contact structure) that is topologically but not Legendrian isotopic to the link Λ1 ⊔ Λ0. We give a complete classification of the Legendrian realizations of this topological link type, as well as all other "cable links" in J1(S1).

Milnor fillable contact structures are universally tight

We show that the canonical contact structure on the link of a normal complex singularity is universally tight. As a corollary we show the existence of closed, oriented, atoroidal 3-manifolds with infinite fundamental groups which carry universally tig ht contact structures that are not deforma- tions of taut (or Reebless) foliations. This answers two questions of Etnyre in (12). Let (X, x) be a normal complex surface singularity. Fix a local embedding of (X, x) in (C N , 0). Then a small sphere S 2N−1 e ⊂ C N centered at the origin intersects X transversely, and the complex hyperplane distribution ξcan on M = X ∩ S 2N−1 e induced by the complex structure on X is called the canonical contact structure. For sufficiently small radius e, the contact manifold is independent of e and the embedding, up to isomorphism. The 3-manifold M is called the link of the singularity, and (M, ξcan) is called the contact boundaryof (X, x). A contact manifold (Y, ξ) is said to be Milnor fillable if it is isomorphic to the contact bound- ary (M, ξcan) of some isolated complex surface singularity (X, x). In addition, we say that a closed and oriented 3-manifold Y is Milnor fillable if it carries a contact structure ξ so that (Y, ξ) is Milnor fillable. It is known that a closed and oriented 3-ma nifold is Milnor fillable if and only if it can be obtained by plumbing according to a weighted graph with negative definite intersection matrix (cf. (25) and (18)). Moreover a ny 3-manifold has at most one Milnor fillable contact structure up to isomorphism(cf. (5)). Note that Milnor fillable contact structures are Stein fillable (see (4)) and hence tight (10). Here we prove that every Milnor fillable contact structure is in fact universally tight, i.e ., the pullback to the universal cover is tight. We would like to point out that universal tightness of a contact structure is not implied by any other type of fillability. In (12), Etnyre settled a question of Eliashberg and Thurston (11) by proving that every contact structure on a closed oriented 3-manifold is obtained by a deformation of a foliation and raised two other related questions:

Central Charge of Supersymmetric 5D Anti–de Sitter Space Solutions of Type IIB Supergravity

We show that generic supersymmetric 5D anti-de Sitter solutions of type IIB supergravity admit a canonical contact structure. This structure determines the central charge of the dual field theory and the conformal dimension of operators dual to supersymmetric wrapped D3-branes. Hence both quantities can be calculated using incomplete information about the solutions, allowing us to prove that they are rational numbers for solutions with a U(1) R symmetry, in agreement with field theory expectations. We also discuss related Duistermaat-Heckman integrals and localization formulas.

The trace problem for Sobolev spaces over the Heisenberg group

This article deals with microlocally weighted Sobolev spaces over the Heisenberg group ℍd, in relation with the canonical subriemannian contact structure $$\kappa = dt + 2(pdq - qdp)$$ . The main purpose is to give a complete description of the restriction to hypersurfaces of functions that belong to those Sobolev spaces, through a trace and lifting theorem. The function spaces of the restrictions involve an additional weight in the space variable, near points where the contact structure of ℍd agrees with the hypersurface.

Smooth s-Cobordisms of Elliptic 3-Manifolds

The main result of this paper states that a symplectic s-cobordism of elliptic 3-manifolds is diffeomorphic to a product (assuming a canonical contact structure on the boundary). Based on this theorem, we conjecture that a smooth s-cobordism of elliptic 3-manifolds is smoothly a product if its universal cover is smoothly a product. We explain how the conjecture fits naturally into the program of Taubes of constructing symplectic structures on an oriented smooth 4-manifold with b+2 ≥ 1 from generic self-dual harmonic forms. The paper also contains an auxiliary result of independent interest, which generalizes Taubes' theorem "SW ⇒ Gr" to the case of symplectic 4-orbifolds.

Singularities of flat fronts in hyperbolic space

It is well-known that the unit cotangent bundle of any Riemannian manifold has a canonical contact structure. A surface in a Riemannian 3-manifold is called a front if it is the projection of a Legendrian immersion into the unit cotangent bundle. We give easily computable criteria for a singular point on a front to be a cuspidal edge or a swallowtail. Using this, we prove that generically flat fronts in hyperbolic 3-space admit only cuspidal edges and swallowtails. We also show that any complete flat front (provided it is not rotationally symmetric) has associated parallel surfaces whose singularities consist of only cuspidal edges and swallowtails.