Search

We prove that the LOSS and GRID invariants of Legendrian links in knot Floer homology behave in certain functorial ways with respect to decomposable Lagrangian cobordisms in the symplectization of the standard contact structure on $\mathbb{R}^3$. Our results give new, computable, and effective obstructions to the existence of such cobordisms., Comment: 28 pages, 16 figures, 1 table

We approach with geometrical tools the contactization and symplectization of filiform structures and define Hamiltonian structures and momentum mappings on Lie groups.

Let $L \subset \mathbb R \times J^1(M)$ be a spin, exact Lagrangian cobordism in the symplectization of the 1-jet space of a smooth manifold $M$. Assume that $L$ has cylindrical Legendrian ends $\Lambda_\pm \subset J^1(M)$. It is well known that the Legendrian contact homology of $\Lambda_\pm$ can be defined with integer coefficients, via a signed count of pseudo-holomorphic disks in the cotangent bundle of $M$. It is also known that this count can be lifted to a mod 2 count of pseudo-holomorphic disks in the symplectization $\mathbb R \times J^1(M)$, and that $L$ induces a morphism between the $\mathbb Z_2$-valued DGA:s of the ends $\Lambda_\pm$ in a functorial way. We prove that this hold with integer coefficients as well. The proofs are built on the technique of orienting the moduli spaces of pseudo-holomorphic disks using capping operators at the Reeb chords. We give an expression for how the DGA:s change if we change the capping operators., Comment: 41 pages, final version, accepted for publication in Quantum Topology. More details have been added to some of the proofs

In this paper we study pseudoholomorphic curves with brake symmetry in symplectization of a closed contact manifold. We introduce the pseudoholomorphic curve with brake symmetry and the corresponding moduli space. Then we get the virtual dimension of the moduli space.

Contact geometry has been successfully employed for the geometric formulation and control of systems containing thermodynamic components. In this paper we elaborate on the geometric theory of symplectization of contact manifolds in order to lift contact control systems to Hamiltonian control systems with a Hamiltonian that is homogeneous in the co-state variables. This provides a new view on contact control systems as used in thermodynamics, and offers possibilities for unifying the theories of contact control systems, Hamiltonian input-output systems and port-Hamiltonian systems.

This is a sequel to the papers Oh and Wang (Real and Complex Submanifolds, Springer Proceedings in Mathematics and Statistics 106 (2014), 43–63, eds. by Y.-J. Suh and et al. for ICM-2014 satellite conference, Daejeon, Korea, August 2014; arXiv:1212.4817; Analysis of contact Cauchy–Riemann maps I: a priori$C^{k}$estimates and asymptotic convergence, submitted, preprint, 2012, arXiv:1212.5186v3). In Oh and Wang (Real and Complex Submanifolds, Springer Proceedings in Mathematics and Statistics 106 (2014), 43–63, eds. by Y.-J. Suh and et al. for ICM-2014 satellite conference, Daejeon, Korea, August 2014; arXiv:1212.4817), the authors introduced a canonical affine connection on $M$ associated to the contact triad $(M,\unicode[STIX]{x1D706},J)$. In Oh and Wang (Analysis of contact Cauchy–Riemann maps I: a priori$C^{k}$estimates and asymptotic convergence, submitted, preprint, 2012, arXiv:1212.5186v3), they used the connection to establish a priori$W^{k,p}$-coercive estimates for maps $w:\dot{\unicode[STIX]{x1D6F4}}\rightarrow M$ satisfying $\overline{\unicode[STIX]{x2202}}^{\unicode[STIX]{x1D70B}}w=0$, $d(w^{\ast }\unicode[STIX]{x1D706}\circ j)=0$without involving symplectization. We call such a pair $(w,j)$ a contact instanton. In this paper, we first prove a canonical neighborhood theorem of the locus $Q$ foliated by closed Reeb orbits of a Morse–Bott contact form. Then using a general framework of the three-interval method, we establish exponential decay estimates for contact instantons $(w,j)$ of the triad $(M,\unicode[STIX]{x1D706},J)$, with $\unicode[STIX]{x1D706}$ a Morse–Bott contact form and $J$ a CR-almost complex structure adapted to $Q$, under the condition that the asymptotic charge of $(w,j)$ at the associated puncture vanishes.We also apply the three-interval method to the symplectization case and provide an alternative approach via tensorial calculations to exponential decay estimates in the Morse–Bott case for the pseudoholomorphic curves on the symplectization of contact manifolds. This was previously established by Bourgeois (A Morse–Bott approach to contact homology, Ph.D. dissertation, Stanford University, 2002) (resp. by Bao (On J-holomorphic curves in almost complex manifolds with asymptotically cylindrical ends, Pacific J. Math. 278(2) (2015), 291–324)), by using special coordinates, for the cylindrical (resp. for the asymptotically cylindrical) ends. The exponential decay result for the Morse–Bott case is an essential ingredient in the setup of the moduli space of pseudoholomorphic curves which plays a central role in contact homology and symplectic field theory (SFT).

Generalizing the canonical symplectization of contact manifolds, we construct an infinite dimensional non-linear Stiefel manifold of weighted embeddings into a contact manifold. This space carries a symplectic structure such that the contact group and the group of reparametrizations act in a Hamiltonian fashion with equivariant moment maps, respectively, giving rise to a dual pair, called the EPContact dual pair. Via symplectic reduction, this dual pair provides a conceptual identification of non-linear Grassmannians of weighted submanifolds with certain coadjoint orbits of the contact group. Moreover, the EPContact dual pair gives rise to singular solutions for the geodesic equation on the group of contact diffeomorphisms. For the projectivized cotangent bundle, the EPContact dual pair is closely related to the EPDiff dual pair due to Holm and Marsden, and leads to a geometric description of some coadjoint orbits of the full diffeomorphism group.

We construct a product on the Floer complex associated to a pair of Lagrangian cobordisms. More precisely, given three exact transverse Lagrangian cobordisms in the symplectization of a contact manifold, we define a map $\mathfrak{m}_2$ by a count of rigid pseudo-holomorphic disks with boundary on the cobordisms and having punctures asymptotic to intersection points and Reeb chords of the negative Legendrian ends of the cobordisms. More generally, to a $(d+1)$-tuple of exact transverse Lagrangian cobordisms we associate a map $\mathfrak{m}_d$ such that the family $(\mathfrak{m}_d)_{d\geq1}$ are $A_\infty$-maps. Finally, we extend the Ekholm-Seidel isomorphism to an $A_\infty$-morphism, giving in particular that it is a ring isomorphism., Comment: 70 pages, 38 figures. Accepted version. Quite a lot of modifications: more rigorous statment of the results, remark 11 turned into a corollary with proof, addition of an example of computation (section 6), modification of notation for some moduli spaces, correction of the definition of (unfinished) pseudo-holomorphic buildings and equivalence relation, section 7 partly rewritten

Eulerian dynamical equations in a three-dimensional domain are used to construct a formal symplectic structure on time-extended space. Symmetries, invariants, and conservation laws are related to this geometric structure. The symplectic structure incorporates dynamics of helicities as identities. The generator of the infinitesimal dilation for symplectic two-form can be interpreted as a current vector for helicity. Symplectic dilation implies the existence of contact hypersurfaces. In particular, these include contact structures on the space of streamlines and on the Bernoulli surfaces.

I define higher codimensional versions of contact structures on manifolds as maximally non-integrable distributions. I call them multicontact structures. Cartan distributions on jet spaces provide canonical examples. More generally, I define higher codimensional versions of pre-contact structures as distributions on manifolds whose characteristic symmetries span a constant dimensional distribution. I call them pre-multicontact structures. Every distribution is almost everywhere, locally, a pre-multicontact structure. After showing that the standard symplectization of contact manifolds generalizes naturally to a (pre-)multisymplectization of (pre-)multicontact manifolds, I make use of results by C. Rogers and M. Zambon to associate a canonical L ∞ -algebra to any (pre-)multicontact structure. Such L ∞ -algebra is a multicontact version of the Jacobi bracket on a contact manifold. Unlike the multisymplectic L ∞ -algebra of Rogers and Zambon, the multicontact L ∞ -algebra is always a homological resolution of a Lie algebra. Finally, I describe in local coordinates the L ∞ -algebra associated to the Cartan distribution on jet spaces.

Consider cotangent bundles of exotic spheres with their canonical symplectic structure. They admit automorphisms that preserve the part at infinity of one fiber and which are analogous to the square of a Dehn twist. Pursuing that analogy, we show that they have infinite order up to isotopy (inside the group of all automorphisms with the same behavior)., Simons Foundation (Simons Investigator Grant), National Science Foundation (U.S.) (Grant DMS-1005288)

In this paper we suggest an approach to the study of the action of point pseudogroup in (contact) 1-jet space J1Rn. This approach is based on the following idea. We replace the canonic projection π1,0:J1Rn→J0(Rn) from 1-jet space to 0-jet space with projective action on the fiber by some bundle, which is called symplectization, with the linear action on the fiber. This makes it possible to apply methods of studying the linear actions of algebraic groups and to find the set of independent differential invariants. Finally, we rewrite the action of point pseudogroup in terms of these invariants and classify the orbits of point pseudogroup action. We also give two examples of application of this method. In the first example we obtain classification of contact vector fields in J1Rn, and in the second we obtain classification of symbols of linear differential operators.

We present a novel [Formula: see text]-characterization of symplectic embeddings and diffeomorphisms in terms of Lagrangian embeddings. Our approach is based on the shape invariant, which was discovered by Sikorav and Eliashberg, intersection theory and the displacement energy of Lagrangian submanifolds, and the fact that non-Lagrangian submanifolds can be displaced immediately. This characterization gives rise to a new proof of [Formula: see text]-rigidity of symplectic embeddings and diffeomorphisms. The various manifestations of Lagrangian rigidity that are used in our arguments come from [Formula: see text]-holomorphic curve methods. An advantage of our techniques is that they can be adapted to a [Formula: see text]-characterization of contact embeddings and diffeomorphisms in terms of coisotropic (or pre-Lagrangian) embeddings, which in turn leads to a proof of [Formula: see text]-rigidity of contact embeddings and diffeomorphisms. We give a detailed treatment of the shape invariants of symplectic and contact manifolds, and demonstrate that shape is often a natural language in symplectic and contact topology. We consider homeomorphisms that preserve shape, and propose a hierarchy of notions of Lagrangian topological submanifold. Moreover, we discuss shape-related necessary and sufficient conditions for symplectic and contact embeddings, and define a symplectic capacity from the shape.

In this paper, we study a Hamiltonian structure of the Vlasov-Poisson system, first mentioned by Fr\"ohlich, Knowles, and Schwarz. To begin with, we give a formal guideline to derive a Hamiltonian on a subspace of complex-valued $L^2$ integrable functions $\alpha$ on the one particle phase space $\mathbb{R}^{2d}$, s.t. $f=\left|\alpha\right|^2$ is a solution of a collisionless Boltzmann equation. The only requirement is a sufficiently regular energy functional on a subspace of distribution functions $f\in L^1$. Secondly, we give a full well-posedness theory for the obtained system corresponding to Vlasov-Poisson in $d\geq3$ dimensions. Finally, we adapt the classical globality results for $d=3$ to the generalized system., Comment: 33 pages, no figures

Geiges and Gonzalo (Invent. Math. 121:147–209 1995, J. Differ. Geom. 46:236–286 1997, Acta. Math. Vietnam 38:145–164 2013) introduced and studied the notion of taut contact circle on a three-manifold. In this paper, we introduce a Riemannian approach to the study of taut contact circles on three-manifolds. We characterize the existence of a taut contact metric circle and of a bi-contact metric structure. Then, we give a complete classification of simply connected three-manifolds which admit a bi-H-contact metric structure. In particular, a simply connected three-manifold admits a homogeneous bi-contact metric structure if and only if it is diffeomorphic to one of the following Lie groups: SU(2), \({\widetilde{SL}}(2,{\mathbb {R}})\), \({\widetilde{E}}(2)\), E(1, 1). Moreover, we obtain a classification of three-manifolds which admit a Cartan structure \((\eta _1,\eta _2)\) with the so-called Webster function \({\mathcal {W}}\) constant along the flow of \(\xi _1\) (equivalently \(\xi _2\)). Finally, we study the metric cone, i.e., the symplectization, of a bi-contact metric three-manifold. In particular, the notion of bi-contact metric structure is related to the notions of conformal symplectic couple (in the sense of Geiges (Duke Math. J. 85:701–711 1996)) and symplectic pair (in the sense of Bande and Kotschick (Trans. Am. Math. Soc. 358(4):1643–1655 2005)).

We study contact structures on nonnegatively-graded manifolds equipped with homological contact vector fields. In the degree 1 case, we show that there is a one-to-one correspondence between such structures (with fixed contact form) and Jacobi manifolds. This correspondence allows us to reinterpret the Poissonization procedure, taking Jacobi manifolds to Poisson manifolds, as a supergeometric version of symplectization., 9 pages. v2: Added references, improved proof of Proposition 3.3. v3: Expanded introduction, clarifying remarks, some changes of sign conventions. Main results are unchanged. v4: Final version, implementing changes suggested by referees

This paper and its prequel (“Part I”) prove a generalization of the usual gluing theorem for two index 1 pseudoholomorphic curves U+ and U− in the symplectization of a contact 3-manifold. We assume that for each embedded Reeb orbit γ, the total multiplicity of the negative ends of U+ at covers of γ agrees with the total multiplicity of the positive ends of U− at covers of γ. However, unlike in the usual gluing story, here the individual multiplicities are allowed to differ. In this situation, one can often glue U+ and U− to an index 2 curve by inserting genus zero branched covers of R-invariant cylinders between them. This paper shows that the signed count of such gluings equals a signed count of zeroes of a certain section of an obstruction bundle over the moduli space of branched covers of the cylinder. Part I obtained a combinatorial formula for the latter count and, assuming the result of the present paper, deduced that the differential ∂ in embedded contact homology satisfies ∂2 = 0. The present paper completes all of the analysis that was needed in Part I. The gluing technique explained here is in principle applicable to more gluing problems. We also prove some lemmas concerning the generic behavior of pseudoholomorphic curves in symplectizations, which may be of independent interest.

We construct a version of rational Symplectic Field Theory for pairs $(X,L)$, where $X$ is an exact symplectic manifold, where $L\subset X$ is an exact Lagrangian submanifold with components subdivided into $k$ subsets, and where both $X$ and $L$ have cylindrical ends. The theory associates to $(X,L)$ a $\Z$-graded chain complex of vector spaces over $\Z_2$, filtered with $k$ filtration levels. The corresponding $k$-level spectral sequence is invariant under deformations of $(X,L)$ and has the following property: if $(X,L)$ is obtained by joining a negative end of a pair $(X',L')$ to a positive end of a pair $(X'',L'')$, then there are natural morphisms from the spectral sequences of $(X',L')$ and of $(X'',L'')$ to the spectral sequence of $(X,L)$. As an application, we show that if $\Lambda\subset Y$ is a Legendrian submanifold of a contact manifold then the spectral sequences associated to $(Y\times\R,\Lambda_k^s\times\R)$, where $Y\times\R$ is the symplectization of $Y$ and where $\Lambda_k^s\subset Y$ is the Legendrian submanifold consisting of $s$ parallel copies of $\Lambda$ subdivided into $k$ subsets, give Legendrian isotopy invariants of $\Lambda$.

We construct almost toric fibrations (ATFs) on all del Pezzo surfaces, endowed with a monotone symplectic form. Except for $\mathbb{C}P^2 \# 1 \overline{\mathbb{C}P^2}$ and $\mathbb{C}P^2 \# 2 \overline{\mathbb{C}P^2}$ , we are able to get almost toric base diagrams (ATBDs) of triangular shape and prove the existence of infinitely many symplectomorphism (in particular Hamiltonian isotopy) classes of monotone Lagrangian tori in $\mathbb{C}P^2 \# k \overline{\mathbb{C}P^2}$, for k=0,3,4,5,6,7,8. We name these tori $\Theta^{n_1,n_2,n_3}_{p,q,r}$. Using the work of Karpov-Nogin, we are able to classify all ATBDs of triangular shape. We are able to prove that $\mathbb{C}P^2 \# 1 \overline{\mathbb{C}P^2}$ also have infinitely many monotone Lagrangian tori up to symplectomorphism and we conjecture that the same holds for $\mathbb{C}P^2 \# 2 \overline{\mathbb{C}P^2}$ . Finally, the Lagrangian tori $\Theta^{n_1,n_2,n_3}_{p,q,r}$ inside a del Pezzo surface $X$ can be seen as monotone fibres of ATFs, such that, over its edge lies a fixed anticanonical symplectic torus $\Sigma$. We argue that $\Theta^{n_1,n_2,n_3}_{p,q,r}$ give rise to infinitely many exact Lagrangian tori in $X \setminus \Sigma$, even after attaching the positive end of a symplectization to the boundary of $X \setminus \Sigma$., Comment: 28 pages, 20 figures

We study pseudoholomorphic maps from a punctured Riemann surface into the symplectization of a contact manifold. A Fredholm theory yields the virtual dimension of the moduli spaces of such maps in terms of the Euler characteristic of the Riemann surface and the asymptotics data given by the periodic solutions of the Reeb vector field associated to the contact form. The transversality results establish the existence of additional structure for these spaces. To be more precise, we prove that these spaces are generically smooth manifolds, and therefore their virtual dimension coincides with their actual dimension.

To any contact manifold one can associate a symplectic manifold called its symplectisation in such a way that contact geometry can be reformulated in terms of equivariant symplectic geometry. Concerning this fundamental construction, a basic question remained open : if two contact manifolds have isomorphic symplectizations, are they isomorphic ? In this thesis, we construct counter-examples to this question. Indeed, in any odd dimension greater than or equal to 5, there exist non-diffeomorphic contact manifolds with isomorphic symplectisations. In addition, we construct two contact structures on a closed manifold that are not conjugate by a diffeomorphism though their symplectizations are isomorphic. The proofs are based on a well-known phenomenon in differential topology (the existence of non-trivial h-cobordisms, detected by Whitehead torsion) as well as flexibility results in symplectic geometry due to Cieliebak and Eliashberg. Another result from this thesis asserts that though these contact manifolds are not isomorphic, they become so after sufficiently many connect sum with a product of spheres., À toute variété de contact, on peut associer canoniquement une variété symplectique appelée sa symplectisation de sorte que la géométrie de contact peut se reformuler en termes de géométrie symplectique équivariante. Au sujet de cette construction fondamentale, une question basique restait ouverte : si deux variété de contact ont des symplectisations isomorphes sont-elles isomorphes ? On construit dans cette thèse des contre-exemples à cette question. Il existe en effet, en toute dimension impaire supérieure ou égale à 5, des variétés de contact non difféomorphes admettant pourtant des symplectisations isomorphes. On construit également, sur une même variété deux structures de contact non conjuguées par un difféomorphisme mais admettant des symplectisations isomorphes. Les démonstrations sont basées sur un phénomène bien connu en topologie différentielle (l'existence de h-cobordismes non triviaux, détectée par la torsion de Whitehead) ainsi que sur des résultats de flexibilité en géométrie symplectique dus à Cieliebak et Eliashberg. Un autre résultat de cette th?e affirme que ces variété de contact, bien que non isomorphes, le deviennent toutefois après un nombre suffisant de sommes connexes avec un produit de sphères.

In any contact manifold of dimension $2n-1\geq 11$, we construct examples of closed legendrian submanifolds which are not diffeomorphic but whose lagrangian cylinders in the symplectization are hamiltonian isotopic., Comment: 8 pages, added an example where the Legendrian submanifolds are smoothly isotopic but not Legendrian isotopic

We give a construction of contact homology in the sense of Eliashberg--Givental--Hofer. Specifically, we construct coherent virtual fundamental cycles on the relevant compactified moduli spaces of pseudo-holomorphic curves., Comment: Final version to appear in JAMS, 102 pages, 10 figures

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

This article describes various moduli spaces of pseudoholomorphic curves on the symplectization of a particular overtwisted contact structure on S 1 x S 2 . This contact structure appears when one considers a closed self dual form on a 4-manifold as a symplectic form on the complement of its zero locus. The article is focussed mainly on disks, cylinders and three-holed spheres, but it also supplies groundwork for a description of moduli spaces of curves with more punctures and non-zero genus.

We construct a differentiable family of non-contactomorphic contact structures on a non-compact (2n-1)-dimensional manifold. This generalizes a result of Eliashberg which shows the existence of non-contactomorphic open solid tori.

We study the problem of the symplectization of a bihamiltonian manifold ( M,Λ0,Λ1 ) and we establish, in the case where M is of odd dimension, the sufficient conditions of this construction in the neighbourhood of a generic point of M . In addition, we give several types of bihamiltonian structures, among which that of Poisson–Nijenhuis, which are locally symplectizable. Finally, we give the sufficient and necessary conditions so that a 2-cobord of the Λ0 -cohomology should be a Poisson tensor.RésuméOn étudie le problème de symplectisation d'une variété bihamiltonienne ( M,Λ0,Λ1 ) et on établit, dans le cas où M est de dimension impaire, les conditions suffisantes de cette construction au voisinage d'un point générique de M . On met ensuite en évidence plusieurs types de structures bihamiltoniennes, entre lesquels celui de Poisson–Nijenhuis, qui sont localement symplectisables. On donne en plus les conditions nécessaires et suffisantes pour que un 2-cobord de la Λ0 -cohomologie soit un tenseur de Poisson.

We provide examples of contact manifolds of any odd dimension greater than or equal to [math] which are not diffeomorphic but have exact symplectomorphic symplectizations.

Given an oriented, compact, 3-dimenional contact manifold (M, λ) we study maps u ˜ = ( a , u ) : ℂ → ℝ × M satisfying the Cauch-Riemann type equation u ˜ s + J ˜ ( u ˜ ) u ˜ t = 0 , with a very special almost complex structure J ˜ related to the contact form λ on M. If the energy is positive and bounded, 0 E ( ( u ˜ ) ∞ , then the asymptotic behavior of u : ℂ → M as |z| → ∞ is intimately related to the dynamics of the Reeb vector field Xλ on M. Assuming the periodic solutions of Xλ to be non degenerate, we shall show that lim R → ∞ u ( R e 2 π i t ) = x ( T t ) for a T-periodic solution x with E ( u ˜ ) = T . The main result is an asymptotic formula which demonstrates the exponential nature of this limit. Some consequences for the geometry of the maps u : ℂ → M are deduced.

We show that the path construction integration of Lie algebroids by Lie groupoids is an actual equivalence from the category of integrable Lie algebroids and complete Lie algebroid comorphisms to the category of source 1-connected Lie groupoids and Lie groupoid comorphisms. This allows us to construct an actual symplectization functor in Poisson geometry. We include examples to show that the integrability of comorphisms and Poisson maps may not hold in the absence of a completeness assumption., Comment: 28 pages, references added

An -structure on a manifold is an endomorphism field satisfying . We call an f-structure regular if the distribution is involutive and regular, in the sense of Palais. We show that when a regular f-structure on a compact manifold M is an almost -structure, it determines a torus fibration of M over a symplectic manifold. When rank , this result reduces to the Boothby-Wang theorem. Unlike similar results for manifolds with -structure or -structure, we do not assume that the f-structure is normal. We also show that given an almost -structure, we obtain an associated Jacobi structure, as well as a notion of symplectization.

We provide simple examples of open manifolds which support inequivalent symplectic structures that are Liouville (exact, and with an infinite end modeled on the convex part of a symplectization)., National Science Foundation (U.S.) (NSF grant DMS-1005288)

Let a torus T act effectively on a compact connected cooriented contact manifold, and let Psi be the natural momentum map on the symplectization. We prove that, if dim T > 2, the union of the origin with the image of Psi is a convex polyhedral cone, the non-zero level sets of Psi are connected (while the zero level set can be disconnected), and the momentum map is open as a map to its image. This answers a question posed by Eugene Lerman, who proved similar results when the zero level set is empty. We also analyze examples with dim T, 39 pages. Contains small corrections and a small simplification of the argument. To appear in Algebraic and Geometric Topology.

Let $(M,\omega)$ be a pseudo-Hermitian space of real dimension $2n+1$, that is $\RManBase$ is a $\CR-$manifold of dimension $2n+1$ and $\omega$ is a contact form on $M$ giving the Levi distribution $HT(M)\subset TM$. Let $M^\omega\subset T^*M$ be the canonical symplectization of $(M,\omega)$ and $M$ be identified with the zero section of $M^\omega$. Then $M^\omega$ is a manifold of real dimension $2(n+1)$ which admit a canonical foliation by surfaces parametrized by $\mathbb{C}\ni t+i\sigma\mapsto \phi_p(t+i\sigma)=\sigma\omega_{g_t(p)}$, where $p\inM$ is arbitrary and $g_t$ is the flow generated by the Reeb vector field associated to the contact form $\omega$. Let $J$ be an (integrable) complex structure defined in a neighbourhood $U$ of $M$ in $M^\omega$. We say that the pair $(U,J)$ is an {adapted complex tube} on $M^\omega$ if all the parametrizations $\phi_p(t+i\sigma)$ defined above are holomorphic on $\phi_p^{-1}(U)$. In this paper we prove that if $(U,J)$ is an adapted complex tube on $M^\omega$, then the real function $E$ on $M^\omega\subset T^*M$ defined by the condition $\alpha=E(\alpha)\omega_{\pi(\alpha)}$, for each $\alpha\in M^\omega$, is a canonical equation for $M$ which satisfies the homogeneous Monge-Amp\`ere equation $(dd^c E)^{n+1}=0$. We also prove that if $M$ and $\omega$ are real analytic then the symplectization $M^\omega$ admits an unique maximal adapted complex tube., Comment: 6 pages

We define the notion of fast finite-energy planes in the symplectization of a closed 3-dimensional energy level $M$ of contact type. We use them to construct special open book decompositions of $M$ when the contact structure is tight and induced by a (non-degenerate) dynamically convex contact form. The obtained open books have disk-like pages that are global surfaces of section for the Hamiltonian dynamics. Let $S \subset \R^4$ be the boundary of a smooth, strictly convex, non-degenerate and bounded domain. We show that a necessary and sufficient condition for a closed Hamiltonian orbit $P\subset S$ to be the boundary of a disk-like global surface of section for the Hamiltonian dynamics is that $P$ is unknotted and has self-linking number -1., 73 pages, some minor corrections made. To appear in Transactions of the American Mathematical Society

We show that every open book decomposition of a contact 3-manifold can be represented (up to isotopy) by a smooth R-invariant family of pseudoholomorphic curves on its symplectization with respect to a suitable stable Hamiltonian structure. In the planar case, this family survives small perturbations, and thus gives a concrete construction of a stable finite energy foliation that has been used in various applications to planar contact manifolds, including the Weinstein conjecture and equivalence of strong and Stein fillability., 13 pages, 2 figures; v3 includes some added details on implicit function theorems for punctured pseudoholomorphic curves; to appear in Expos. Math

A symplectic manifold $W$ with contact type boundary $M = \partial W$ induces a linearization of the contact homology of $M$ with corresponding linearized contact homology $HC(M)$. We establish a Gysin-type exact sequence in which the symplectic homology $SH(W)$ of $W$ maps to $HC(M)$, which in turn maps to $HC(M)$, by a map of degree -2, which then maps to $SH(W)$. Furthermore, we give a description of the degree -2 map in terms of rational holomorphic curves with constrained asymptotic markers, in the symplectization of $M$., Comment: Final version. Changes for v2: Proof of main theorem supplemented with detailed discussion of continuation maps. Description of degree -2 map rewritten with emphasis on asymptotic markers. Sec. 5.2 rewritten with emphasis on 0-dim. moduli spaces. Transversality discussion reorganized for clarity (now Remark 9). Various other minor modifications

The embedded contact homology (ECH) of a 3-manifold with a contact form is a variant of Eliashberg-Givental-Hofer's symplectic field theory, which counts certain embedded J-holomorphic curves in the symplectization. We show that the ECH of T^3 is computed by a combinatorial chain complex which is generated by labeled convex polygons in the plane with vertices at lattice points, and whose differential involves `rounding corners'. We compute the homology of this combinatorial chain complex. The answer agrees with the Ozsvath--Szabo Floer homology HF^+(T^3)., This is the version published by Geometry & Topology on 26 March 2006

It is well-known that all problems of Contact geometry can be reformulated as problems of Symplectic geometry. This can be done via symplectization (see 2.1 below). In particular, the problem of Lagrangian intersections naturally arises in connection with several contact geometric questions (see 2.5 example, and below). However, there is one major difficulty when one tries to realize this approach: the symplectizations of contact manifolds are noncompact and, what is even worse, non-convex (see [EGr1]). This leads to the loss of compactness for the spaces of holomorphic curves and thus creates serious difficulties for the traditional Floer homology approach. The goal of this paper is to show that this problem can be successfully overcome by using an idea from [H].