Reeb vector fields and open book decompositions math

Colin reeb vector fields and open book decompositions, j. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. 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. Positive topological entropy for reeb flows on 3dimensional. 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. 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. The relation between geodesible vector fields and open book decompositions was already suggested. Reeb vector fields and open book decompositions math. For graphics coders, vector and matrix math libraries are something we use nearly every day, and in just about every function we write.

Presas, foliated vector fields without periodic orbits, isr. Ko honda usc reeb vector fields june 11, 2007 12 25. The argument is made using properties of differential forms, in the context of open book decompositions of contact 3manifolds. Foote math 225 recall our main theorem about vector. These parallel the descriptions in and in the appendix. 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. 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. Topological entropy for reeb vector fields in dimension. As applications, we quantitatively study the gap between overtwisted contact structures and nonrightveering monodromies. Abstract a contact structure is a smooth distribution of hyperplanes on an odddimensional manifold that is nonintegrable everywhere. The relation between geodesible vector fields and open book decompositions was.

Feg luo, rutgers university volume and angle structures on 3manifolds. Eleny ionel, stanford university singular spaces and gromovwitten invariants. In mathematics, floer homology is a tool for studying symplectic geometry and lowdimensional topology. Andreas floer introduced the first version of floer homology, now called lagrangian floer homology, in his proof of the arnold conjecture in symplectic geometry. We develop a variant of lusternikschnirelmann theory for the shift operator in equivariant floer and symplectic homology. Reeb vector fields and open book decompositions arxiv. The generalized svd decomposition is usually written as. A contact structure on mis a maximally nonintegrable hyperplane eld tm. The argument is made using properties of differential forms, in the context of open book decompositions of contact 3. In other words, lx g for some function g, where lis the lie derivative. Conan leung chinese university of hong kong geometry of special holonomy april 16. 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. For much more about this see bourgeoisekholmeliashberg, effect of legendrian surgery.

In section 2 we recall the definitions and properties of open books, fillability, the heegaard floer contact. Download citation reeb vector fields and open book decompositions we determine parts of the contact homology of certain contact 3manifolds in the framework of. The two matrices must have the same number of rows. We compute parts of the contact homology of contact 3manifolds which are supported by open books with pseudoanosov monodromy. In the case of an open book with pseudoanosov monodromy, suppose the. Given an open book decomposition of a closed contact three manifold m. Discussing the handle decompositions and induced stein. Tight planar contact manifolds with vanishing heegaard. Zoltan szabo princeton university link floer homology and the. Reeb vector fields and open book decompositions 3 were obtained by yau y2, y3. William meeks university of massachusetts the dynamics theorem for embedded minimal surfaces. Pullback attractors for a singularly nonautonomous plate equation.

Topological entropy for reeb vector fields in dimension three. A constructive method to carry out one of the decompositions is applied to some three. An open book decomposition of the tight 3sphere with one binding orbit and disklike pages, which are transverse to the reeb vector field. Such techniques have the potential for applications as foundations for invariants in symplectic topology arising from. Sep 29, 2008 we determine parts of the contact homology of certain contact 3manifolds in the framework of open book decompositions, due to giroux. Twoday conference on probabilistic combinatorics and algorithms. Floer homology is a novel invariant that arises as an infinitedimensional analogue of finitedimensional morse homology. Research of ko honda ucla department of mathematics.

We develop techniques for defining and working with virtual fundamental cycles on moduli spaces of pseudoholomorphic curves which are not necessarily cut out transversally. Tight planar contact manifolds with vanishing heegaard floer. Topological entropy for reeb vector fields in dimension three via open book decompositions authors. Legendrian contact homology and topological entropy journal. On the sign of the curvature of a contact metric manifold. Yes, the reeb field rw associated to a contact form w satisfies. Positive topological entropy for reeb flows on 3dimensional anosov contact manifolds. It 202042 processes associated with ionic current rectification at a 2dtitanate nanosheet deposit on a microhole poly ethylene terephthalate substrate. M is a link called the binding, and the fibers are called.

A geometric obstruction, the so called plastikstufe, for a contact structure to not being fillable has been found in kl06. What enables us to get a handle on the contact homology is a better understanding of tightness in the open book framework. Previous geometry festival speakers new york university. V colin, k honda, reeb vector fields and open book decompositions arxiv. The subject of this note is a correspondence between open book decompositions on contact manifolds and jholomorphic curves in their symplectizations.

Alexei borodin caltech random surfaces in dimension. For an application see colinhonda, reeb vector fields and open book decompositions. Symplectic floer homology sfh is a homology theory associated to a symplectic manifold and a nondegenerate symplectomorphism of it. Topological entropy for reeb vector fields in dimension three via open book decompositions marcelo r. Reeb vector fields and open book decompositions core. This paper elaborates on the theory showing a big number of closed contact manifolds having a plastikstufe. Mathematics geometric topology, mathematics differential geometry, 57m50, 53c15. Positive topological entropy of reeb flows on spherizations volume 151 issue 1 leonardo macarini, felix schlenk. Yair minsky yale university asymptotic geometry of the mapping class group.

So, they are the first examples of nonfillable high. Abstractlet m3, g be an almost kenmotsu 3manifold such that the reeb vector field is an eigenvector field of the ricci operator. Hence, the study of contact structures reduces to the study of monodromy maps h. M thomas vogel, iasupenn existence of engel structures. An algebraic approach to virtual fundamental cycles on moduli spaces of pseudoholomorphic curves. Open book decompositions and stable hamiltonian structures. A fundamental result of giroux gi is that every contact structure on a closed 3manifold is supported by some open book decomposition. S is an oriented compact surface with boundary called the page of the open book decomposition. Essential open book foliations and fractional dehn twist. 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. Our key result is that the spectral invariants are strictly decreasing under the action of the shift operator when periodic orbits are isolated. For an open book with periodic monodromy, we verify the weinstein conjecture. Then the nonintegrability condition is equivalent to d n0. 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.

We determine parts of the contact homology of certain contact 3manifolds in the framework of open book decompositions, due to giroux. Frank morgan williams college manifolds with density. Let r be an open region in e2 and let f be a c1 vector. A curlfree vector field that is not a gradient robert l. Notice that a reeb field exists by linear algebra alone. 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.

Finding reeb vector fields associated with contact forms. Suppose that there exists a global 1form such that. Abstract we determine parts of the contact homology of certain contact 3manifolds in the framework of open book decompositions, due to giroux. Roughly speaking, rightveering means it sends all properly embedded arcs on s to the right. In this paper we prove that, for large classes of tight contact 3manifolds, hcm. Some theorems are proved concerning the decomposition of vector fields into gradient and hamiltonian components. We discuss spinal open books and the natural contact structures we associate to them in section 3. 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. Jul 12, 2011 on vector fields having properties of reeb fields. Tuesday seminar on topology graduate school of mathematical. If a contact manifold arises as a constantenergy hypersurface inside a symplectic manifold, then the reeb vector field is the restriction to the. Geometric decompositions of almost contact manifolds contact and symplectic topology, 7172, bolyai soc.

Ko honda usc reeb vector fields june 11, 2007 11 25. V colin, k honda, reeb vector fields and open book decompositions, j. Mikhail kovanov, columbia university braid cobordisms and triangulated categories. Legendrian contact homology and topological entropy. In this expository article, we discuss the authors conjecture that an associated metric for a given contact form on a contact manifold of dimension. This means the reeb field lives in the tangent space to the link, along the positive direction, and r. Families of contact 3manifolds with arbitrarily large. Lusternikschnirelmann theory and closed reeb orbits. As an application, we prove new multiplicity results for simple closed reeb orbits on the standard contact sphere, the unit.