Dimensional Contact Manifolds Research Articles

On contact invariants of non-simply connected Gorenstein toric contact manifolds

The first two authors showed in~\cite{AM1} how the Conley-Zehnder index of any contractible periodic Reeb orbit of a non-degenerate toric contact form on a good toric contact manifold with zero first Chern class, i.e. a Gorenstein toric contact manifold, can be explicitly computed using moment map data. In this paper we show that the same explicit method can be used to compute Conley-Zehnder indices of non-contractible periodic Reeb orbits. Under appropriate conditions, the (finite) number of such orbits in a given free homotopy class and with a given index is a contact invariant of the underlying contact manifold. We compute these invariants for two sets of examples that satisfy these conditions: $5$-dimensional contact manifolds that arise as unit cosphere bundles of $3$-dimensional lens spaces, and $2n+1$-dimensonal Gorenstein contact lens spaces. As illustrative contact topology applications, we show that diffeomorphic lens spaces might not be contactomorphic and that there are homotopy classes of diffeomorphisms of some lens spaces that do not contain any contactomorphism.

SFT COMPUTATIONS AND INTERSECTION THEORY IN HIGHER-DIMENSIONAL CONTACT MANIFOLDS

AbstractI construct infinitely many nondiffeomorphic examples of $5$ -dimensional contact manifolds which are tight, admit no strong fillings and do not have Giroux torsion. I obtain obstruction results for symplectic cobordisms, for which I give a proof not relying on the polyfold abstract perturbation scheme for Symplectic Field Theory (SFT). These results are part of my PhD thesis [23], and are the first applications of higher-dimensional Siefring intersection theory for holomorphic curves and hypersurfaces, as outlined in [23, 24], as a prequel to [30].

Examples of contact mapping classes of infinite order in all dimensions

We give examples of tight high dimensional contact manifolds admitting a contactomorphism whose powers are all smoothly isotopic but not contact-isotopic to the identity. This is a generalization of an observation in dimension 3 by Gompf, also reused by Ding and Geiges.

Pseudo Cauchy Riemann and Framed Manifolds with Physical Applications

We introduce a pseudo Cauchy Riemann(PCR)-structure defined by a real tensor field $\bar{J}$ of type $(1, 1)$ of a real semi-Riemannian manifold $(\bar{M}, \bar{g})$ such that $\bar{J}^2 = \lambda^2 I$, where $\lambda$ is a function on $\bar{M}$. We prove that, contrary to the even dimensional CR-manifolds, a PCR-manifold is not necessarily of even dimension if $\lambda$ is every where non-zero real function on $\bar{M}$, supported by two odd dimensional examples and one physical model. The metric of PCR-manifold is not severely restricted. Then, we define a pseudo framed(PF)-manifold $(M, g)$ by a real tensor field $f$ such that $f^3 = \lambda^2 f$, where $T(M)$ splits into a direct sum of two subbundles, namely $im(f)$ (with a PCR-structure) and $ ker(f)$, supported by some mathematical and physical examples. Finally, we study a revised version of a contact manifold, called contact PF-manifold, which is a particular case of a PF-manifold where dim$(ker(f))=1$. Contrary to the odd dimensional contact manifolds, there do exist even dimensional contact PF-manifolds. We also propose several open problems.

On some examples and constructions of contact manifolds

The first goal of this paper is to construct examples of higher dimensional contact manifolds with specific properties. Our main results in this direction are the existence of tight virtually overtwisted closed contact manifolds in all dimensions and the fact that every closed contact 3-manifold, which is not (smoothly) a rational homology sphere, contact-embeds with trivial normal bundle inside a hypertight closed contact 5-manifold. This uses known construction procedures by Bourgeois (on products with tori) and Geiges (on branched covering spaces). We pass from these procedures to definitions; this allows to prove a uniqueness statement in the case of contact branched coverings, and to study the global properties (such as tightness and fillability) of the results of both constructions without relying on any auxiliary choice in the procedures. A second goal allowed by these definitions is to study relations between these constructions and the notions of supporting open book, as introduced by Giroux, and of contact fiber bundle, as introduced by Lerman. For instance, we give a definition of Bourgeois contact structures on flat contact fiber bundles which is local, (strictly) includes the results of the Bourgeois construction, and allows to recover an isotopy class of supporting open books on the fibers. This last point relies on a reinterpretation, inspired by an idea by Giroux, of supporting open books in terms of pairs of contact vector fields.

Contact manifolds, Lagrangian Grassmannians and PDEs

Abstract In this paper we review a geometric approach to PDEs. We mainly focus on scalar PDEs in n independent variables and one dependent variable of order one and two, by insisting on the underlying (2n + 1)-dimensional contact manifold and the so-called Lagrangian Grassmannian bundle over the latter. This work is based on a Ph.D course given by two of the authors (G. M. and G. M.). As such, it was mainly designed as a quick introduction to the subject for graduate students. But also the more demanding reader will be gratified, thanks to the frequent references to current research topics and glimpses of higher-level mathematics, found mostly in the last sections.

A categorification of the Alexander polynomial in embedded contact homology

Given a transverse knot $K$ in a three dimensional contact manifold $(Y,\alpha)$, in [13] Colin, Ghiggini, Honda and Hutchings define a hat version of embedded contact homology for $K$, that we call $\widehat{ECK}(K,Y,\alpha)$, and conjecture that it is isomorphic to the knot Floer homology $\widehat{HFK}(K,Y)$. We define here a full version $ECK(K,Y,\alpha)$ and generalise the definitions to the case of links. We prove then that, if $Y = S^3$, $ECK$ and $\widehat{ECK}$ categorify the (multivariable) Alexander polynomial of knots and links, obtaining expressions analogue to that for knot and link Floer homologies in the plus and, respectively, hat versions.

The topology of Stein fillable manifolds in high dimensions, II

We continue our study of contact structures on manifolds of dimension at least five using complex surgery theory. We show that in each dimension 2q+1 > 3 there are 'maximal' almost contact manifolds to which there is a Stein cobordism from any other (2q+1)-dimensional contact manifold. We show that the product M x S^2 admits a weakly fillable contact structure provided M admits a weak symplectic filling. We also study the connection between Stein fillability and connected sums: we give examples of almost contact manifolds for which the connected sum is Stein fillable, while the components are not. Concerning obstructions to Stein fillings, we show that the (8k-1)-dimensional sphere has an almost contact structure which is not Stein fillable once k > 1. As a consequence we deduce that any highly connected almost contact (8k-1)-manifold (with k > 1) admits an almost contact structure which is not Stein fillable. The proofs rely on a new number-theoretic result about Bernoulli numbers.

An exact sequence for Legendrian links

A result of Bourgeois, Ekholm and Eliashberg [4] describes the linearized contact homology of the boundary of a symplectic cobordism obtained by Legendrian surgery in terms of the cyclic homology of the composable part of the Legendrian homology algebra of the Legendrian attaching spheres. In this work we consider the case when the attaching locus has multiple components, describe the relevant version of the homology algebra of the Legendrian attaching spheres remaining after surgery along one sphere, and we obtain an exact sequence for the original link. Let us recall some basic concepts from contact topology. Let Y be a .2n 1/‐ dimensional contact manifold equipped with a maximally nonintegrable field of tangent hyperplanes . A Legendrian submanifold E Y is an .n 1/‐submanifold of Y which is tangent to everywhere. Weinstein’s description of contact surgery on Y along a Legendrian sphere E Y gives a new contact manifold YE and a symplectic cobordism X from Y to YE ; see Weinstein [20]. In this construction one performs surgery along the Legendrian sphere E with trivial conformal symplectic normal bundle in the contact manifold Y . The resulting manifold YE carries again a contact structure which coincides with the old one outside the neighborhood where surgery takes place. The symplectic cobordism between the contact manifolds Y and YE can be written as the union of I Y (where I is an interval) and a standard handle which is embedded in R 2n and has the standard symplectic structure; see [20] for an explicit model of the handle and the attaching procedure. The product I Y has a symplectic structure as part of the symplectization of Y . Using a normal form for neighborhoods of Legendrian submanifolds in contact

Fredholm pseudo-gradients for the action functional on a sub-manifold of dual Legendrian curves of a three dimensional contact manifold (M3, α)

We prove in this paper that the intersection numbers between periodic orbits have an intrinsic meaning for the variational problem (J,C β ) {Bahri (Pseudo-Orbits of Contact Forms Pitman Research Notes in Mathematics Series No. 173, 1984), Bahri (C R Acad Sci Paris 299, Serie I 15:757–760, 1984), Bahri (Classical and Quantic periodic motions of multiply polarized spin-manifolds. Pitman Research Notes in Mathematics Series No. 378, 1998)}, corresponding to the periodic orbit problem on a sub-manifold of the loop space of a three dimensional compact contact manifold (M, α).

Moment-Angle Manifolds, Intersection of Quadrics and Higher Dimensional Contact Manifolds

We construct new examples of contact manifolds in arbitrarily large dimensions. These manifolds which we call quasi moment-angle manifolds, are closely related to the classical moment-angle manifolds.

The sharp lower bound of the first eigenvalue of the sub-Laplacian on a quaternionic contact manifold in dimension seven

A version of Lichnerowicz’ theorem giving a lower bound of the eigenvalues of the sub-Laplacian on a compact seven dimensional quaternionic contact manifold is proved assuming a lower bound on the Sp(1)Sp(1)-components of the qc-Ricci curvature and the positivity of the P-function of any eigenfunction. The introduced P-function and nonlinearC-operator are motivated by the Paneitz operators defined previously in the Riemannian and CR settings and the P-function used in the theory of elliptic partial differential equations. It is shown that in the case of a seven dimensional compact 3-Sasakian manifold the lower bound is reached iff the quaternionic contact manifold is the round 3-Sasakian sphere.

On Surfaces in Three Dimensional Contact Manifolds

AbstractIn this paper, we introduce two quantities defined on a surface in a contact manifold. The first one is called degree of transversality (DOT), which measures the transversality between the tangent spaces of a surface and the contact planes. The second quantity, called curvature of transversality (COT), is designed to give a comparison principle for DOT along characteristic curves under bounds on COT. In particular, this gives estimates on lengths of characteristic curves, assuming COT is bounded below by a positive constant. We show that surfaces with constant COT exist, and we classify all graphs in the Heisenberg group with vanishing COT. This is accomplished by showing that the equation for graphs with zero COT can be decomposed into two first order PDEs, one of which is the backward invisicid Burgers’ equation. Finally we show that the p-minimal graph equation in the Heisenberg group also has such a decomposition. Moreover, we can use this decomposition to write down an explicit formula of a solution near a regular point.

Weak and strong fillability of higher dimensional contact manifolds

For contact manifolds in dimension three, the notions of weak and strong symplectic fillability and tightness are all known to be inequivalent. We extend these facts to higher dimensions: in particular, we define a natural generalization of weak fillings and prove that it is indeed weaker (at least in dimension five), while also being obstructed by all known manifestations of “overtwistedness”. We also find the first examples of contact manifolds in all dimensions that are not symplectically fillable but also cannot be called overtwisted in any reasonable sense. These depend on a higher dimensional analogue of Giroux torsion, which we define via the existence in all dimensions of exact symplectic manifolds with disconnected contact boundary.

Twisted supersymmetric 5D Yang-Mills theory and contact geometry

We extend the localization calculation of the 3D Chern-Simons partition func- tion over Seifert manifolds to an analogous calculation in five dimensions. We construct a twisted version of N = 1 supersymmetric Yang-Mills theory defined on a circle bundle over a four dimensional symplectic manifold. The notion of contact geometry plays a crucial role in the construction and we suggest a generalization of the instanton equations to five- dimensional contact manifolds. Our main result is a calculation of the full perturbative partition function on S 5 for the twisted supersymmetric Yang-Mills theory with different Chern-Simons couplings. The final answer is given in terms of a matrix model. Our construction admits generalizations to higher dimensional contact manifolds. This work is inspired by the work of Baulieu-Losev-Nekrasov from the mid 90’s, and in a way it is covariantization of their ideas for a contact manifold.

Contact geometry of multidimensional Monge-Ampère equations: characteristics, intermediate integrals and solutions

Nous étudions la géométrie des équations aux dérivées partielles scalaires du deuxième ordre multidimensionnelles (c’est-à-dire, EDP avec n variables indépendantes), considérées comme hypersurfaces ℰ dans le fibré Grassmannien Lagrangien M (1) sur une variété de contact (2n+1)-dimensionnelle (M,𝒞). Nous développons la théorie des caractéristiques de ℰ en termes de la géométrie de contact et de la géométrie du fibré Grassmannien Lagrangien et étudions leur relation avec les intégrales intermédiaires de ℰ. Après avoir appliqué tels résultats aux équations de Monge-Ampère générales (EMA), nous concentrons notre attention sur les EMA du type introduit par Goursat en 1899 :

Tightness in contact metric 3-manifolds

This paper begins the study of relations between Riemannian geometry and global properties of contact structures on 3-manifolds. In particular we prove an analog of the sphere theorem from Riemannian geometry in the setting of contact geometry. Specifically, if a given three dimensional contact manifold (M,ξ) admits a complete compatible Riemannian metric of positive 4/9-pinched curvature then the underlying contact structure ξ is tight; in particular, the contact structure pulled back to the universal cover is the standard contact structure on S 3. We also describe geometric conditions in dimension three for ξ to be universally tight in the nonpositive curvature setting.

Branch structure of 𝐽–holomorphic curves near periodic orbits of a contact manifold

Let M M be a three–dimensional contact manifold, and ψ ~ : D ∖ { 0 } → M × R \tilde {\psi }:D\setminus \{0\}\to M\times {\mathbb R} a finite–energy pseudoholomorphic map from the punctured disc in C {\mathbb C} that is asymptotic to a periodic orbit of the contact form. This article examines conditions under which smooth coordinates may be defined in a tubular neighbourhood of the orbit such that ψ ~ \tilde {\psi } resembles a holomorphic curve, invoking comparison with the theory of topological linking of plane complex algebroid curves near a singular point. Examples of this behaviour, which are studied in some detail, include pseudoholomorphic maps into E p , q × R {\mathbb E}_{p,q}\times {\mathbb R} , where E p , q {\mathbb E}_{p,q} denotes a rational ellipsoid (contact structure induced by the standard complex structure on C 2 {\mathbb C}^{2} ), as well as contact structures arising from non-standard circle–fibrations of the three–sphere.

Linking, Legendrian linking and causality

The set N of all null geodesics of a globally hyperbolic (d + 1)-dimensional spacetime (M, g) is naturally a smooth (2d − 1)-dimensional contact manifold. The sky of an event x in M is the subset X of N consisting of all null geodesics through x, and is an embedded Legendrian submanifold of N diffeomorphic to S(d − 1). It was conjectured by Low that for d = 2 two events x and y are causally related if and only if X and Y are linked (in an appropriate sense). We use the contact structure and knot polynomial calculations to prove this conjecture in certain particular cases, and suggest that for d = 3 smooth linking should be replaced with Legendrian linking.

Pseudoholomorphic strips in symplectizations II: Fredholm theory and transversality

AbstractThis paper is part of a larger program, the investigation of the chord problem in three dimensional contact geometry. The main tool will be pseudoholomorphic strips in the symplectisation of a three dimensional contact manifold with two totally real submanifolds L0, L1 as boundary conditions. The submanifolds L0 and L1 do not intersect transversally. In this paper we will develop a nonlinear Fredholm theory that guarantees the existence of a family of embedded pseudoholomorphic strips near a given one with suitable properties. © 2003 Wiley Periodicals, Inc.