Reeb vector fields and open book decompositions math

If the symplectomorphism is hamiltonian, the homology arises from studying the symplectic action functional on the universal cover of the free loop space of a symplectic manifold. Suppose that there exists a global 1form such that. The two matrices must have the same number of rows. Feg luo, rutgers university volume and angle structures on 3manifolds. Reeb vector fields and open book decompositions math. On the sign of the curvature of a contact metric manifold. The question is about the last sentence of the last corollary of stabilizing the monodromy of an open book decomposition by vicent colin and ko honda. A geometric obstruction, the so called plastikstufe, for a contact structure to not being fillable has been found in kl06. Reeb vector fields and open book decompositions arxiv. Pullback attractors for a singularly nonautonomous plate equation. A constructive method to carry out one of the decompositions is applied to some three.

We introduce essential open book foliations by refining open book foliations, and develop technical estimates of the fractional dehn twist coefficient fdtc of monodromies and the fdtc for closed braids, which we introduce as well. Alexei borodin caltech random surfaces in dimension two, three and four april 30 mihalis dafermos m i t. In mathematics, floer homology is a tool for studying symplectic geometry and lowdimensional topology. Download citation reeb vector fields and open book decompositions we determine parts of the contact homology of certain contact 3manifolds in the framework of open book decompositions, due to. This generalizes somehow the concept of overtwisted structure to dimensions higher than 3. The subject of this note is a correspondence between open book decompositions on contact manifolds and jholomorphic curves in their symplectizations. Ko honda usc reeb vector fields june 11, 2007 11 25. Zoltan szabo princeton university link floer homology and the.

V colin, k honda, reeb vector fields and open book decompositions, j. Abstract we determine parts of the contact homology of certain contact 3manifolds in the framework of open book decompositions, due to giroux. A curlfree vector field that is not a gradient robert l. Presas, foliated vector fields without periodic orbits, isr. William meeks university of massachusetts the dynamics theorem for embedded minimal surfaces. Jul 12, 2011 on vector fields having properties of reeb fields. In this paper, we prove that if g represents a ricci soliton whose potential vector field is orthogonal to the reeb vector field, then m3 is locally isometric to either the hyperbolic space. For graphics coders, vector and matrix math libraries are something we use nearly every day, and in just about every function we write. Given an open book decomposition of a closed contact three manifold m. As an application, we prove new multiplicity results for simple closed reeb orbits on the standard contact sphere, the unit. Legendrian contact homology and topological entropy. As applications, we quantitatively study the gap between overtwisted contact structures and nonrightveering monodromies. In this expository article, we discuss the authors conjecture that an associated metric for a given contact form on a contact manifold of dimension.

Mikhail kovanov, columbia university braid cobordisms and triangulated categories. For an application see colinhonda, reeb vector fields and open book decompositions. M thomas vogel, iasupenn existence of engel structures. The generalized svd decomposition is usually written as. Tight planar contact manifolds with vanishing heegaard. We discuss spinal open books and the natural contact structures we associate to them in section 3. The relation between geodesible vector fields and open book decompositions was. Topological entropy for reeb vector fields in dimension. This paper elaborates on the theory showing a big number of closed contact manifolds having a plastikstufe.

Frank morgan williams college manifolds with density. What enables us to get a handle on the contact homology is a better understanding of tightness in the open book framework. Ko honda, reeb vector fields and open book decompositions william meeks, the dynamics theorem for embedded minimal surfaces yair minsky, asymptotic geometry of the mapping class group frank morgan, manifolds with density zoltan szabo, link floer homology and the thurston norm 2007 at maryland. The relation between geodesible vector fields and open book decompositions was already suggested. Research of ko honda ucla department of mathematics. Hopefully, too, someone could help me figure out how to show that a contact vector field one whose flow preserves the contact structure, that is transverse to the contact pages is a reeb field.

Abstractlet m3, g be an almost kenmotsu 3manifold such that the reeb vector field is an eigenvector field of the ricci operator. Tight planar contact manifolds with vanishing heegaard floer. Such techniques have the potential for applications as foundations for invariants in symplectic topology arising from. In section 2 we recall the definitions and properties of open books, fillability, the heegaard floer contact. Yair minsky yale university asymptotic geometry of the mapping class group.

Given an open book decomposition of a contact three manifold m. Hence, the study of contact structures reduces to the study of monodromy maps h. We compute parts of the contact homology of contact 3manifolds which are supported by open books with pseudoanosov monodromy. Colin reeb vector fields and open book decompositions, j. Hopefully, too, someone could help me figure out how to show that a contact vector field one whose flow preserves the contact structure, that is transverse to the contact. An algebraic approach to virtual fundamental cycles on moduli spaces of pseudoholomorphic curves. We determine parts of the contact homology of certain contact 3manifolds in the framework of open book decompositions, due to giroux. M is a link called the binding, and the fibers are called. Reeb vector fields and open book decompositions ko honda university of southern california june 11, 2007 joint work with vincent colin ko honda usc reeb vector fields june 11, 2007 1 25. Some theorems are proved concerning the decomposition of vector fields into gradient and hamiltonian components. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. Tuesday seminar on topology graduate school of mathematical.

In the case of dimension 3, there is a nice relationship between open book decompositions of 3manifolds and contact structures up to contactomorphisms, called giroux correspondence. Eleny ionel, stanford university singular spaces and gromovwitten invariants. Topological entropy for reeb vector fields in dimension three via open book decompositions marcelo r. Mathematics geometric topology, mathematics differential geometry, 57m50, 53c15. Essential open book foliations and fractional dehn twist. Families of contact 3manifolds with arbitrarily large. A fundamental result of giroux gi is that every contact structure on a closed 3manifold is supported by some open book decomposition. Conan leung chinese university of hong kong geometry of special holonomy april 16. Twoday conference on probabilistic combinatorics and algorithms.

Let r be an open region in e2 and let f be a c1 vector. Previous geometry festival speakers new york university. Roughly speaking, rightveering means it sends all properly embedded arcs on s to the right. The argument is made using properties of differential forms, in the context of open book decompositions of contact 3. We develop a variant of lusternikschnirelmann theory for the shift operator in equivariant floer and symplectic homology. In this paper we prove that, for large classes of tight contact 3manifolds, hcm. So, they are the first examples of nonfillable high. A contact structure on mis a maximally nonintegrable hyperplane eld tm.

Download citation reeb vector fields and open book decompositions we determine parts of the contact homology of certain contact 3manifolds in the framework of. Ko honda usc reeb vector fields june 11, 2007 12 25. Foote math 225 recall our main theorem about vector. Abstract a contact structure is a smooth distribution of hyperplanes on an odddimensional manifold that is nonintegrable everywhere. Positive topological entropy of reeb flows on spherizations volume 151 issue 1 leonardo macarini, felix schlenk. Lusternikschnirelmann theory and closed reeb orbits. If a contact manifold arises as a constantenergy hypersurface inside a symplectic manifold, then the reeb vector field is the restriction to the. Finding reeb vector fields associated with contact forms. Yes, the reeb field rw associated to a contact form w satisfies. Open book decompositions and stable hamiltonian structures. The argument is made using properties of differential forms, in the context of open book decompositions of contact 3manifolds. For completeness, we show, using convex surface theory, that there is a unique choice of a compatible contact structure on a given spinal open book propositions 3. Ko honda, reeb vector fields and open book decompositions william meeks, the dynamics theorem for embedded minimal surfaces yair minsky, asymptotic geometry of the mapping class group frank morgan, manifolds with density zoltan szabo, link floer homology and.

Topological entropy for reeb vector fields in dimension three via open book decompositions authors. Symplectic floer homology sfh is a homology theory associated to a symplectic manifold and a nondegenerate symplectomorphism of it. Then the nonintegrability condition is equivalent to d n0. Discussing the handle decompositions and induced stein. In the case of an open book with pseudoanosov monodromy, suppose the. Notice that a reeb field exists by linear algebra alone. Alexei borodin caltech random surfaces in dimension. It 202042 processes associated with ionic current rectification at a 2dtitanate nanosheet deposit on a microhole poly ethylene terephthalate substrate. The generalized singular value decomposition or gsvd of a pair of matrices a and b rewrites each matrix as the product of an orthogonal or unitary matrix, a diagonal matrix, and the product of the same triangular and orthogonal matrix. For an open book with periodic monodromy, we verify the weinstein conjecture. V colin, k honda, reeb vector fields and open book decompositions arxiv. In other words, lx g for some function g, where lis the lie derivative. The generalized singular value decomposition matrix.

An open book decomposition of the tight 3sphere with one binding orbit and disklike pages, which are transverse to the reeb vector field. A few years ago, cliff taubes proved the weinstein conjecture which says that every reeb vector field on a closed, oriented 3manifold has a closed aka periodic orbit. In dimension 3, the standard contact structure on the 3torus admits a flat associated metric. We develop techniques for defining and working with virtual fundamental cycles on moduli spaces of pseudoholomorphic curves which are not necessarily cut out transversally. Our key result is that the spectral invariants are strictly decreasing under the action of the shift operator when periodic orbits are isolated. Reeb vector fields and open book decompositions core. Positive topological entropy for reeb flows on 3dimensional anosov contact manifolds. These parallel the descriptions in and in the appendix. For much more about this see bourgeoisekholmeliashberg, effect of legendrian surgery.

This means the reeb field lives in the tangent space to the link, along the positive direction, and r. S is an oriented compact surface with boundary called the page of the open book decomposition. Geometric decompositions of almost contact manifolds contact and symplectic topology, 7172, bolyai soc. Reeb vector fields and open book decompositions 3 were obtained by yau y2, y3. Pdf on vector fields having properties of reeb fields. Topological entropy for reeb vector fields in dimension three.