Reeb vector fields and open book decompositions mathematics

We develop a variant of lusternikschnirelmann theory for the shift operator in equivariant floer and symplectic homology. Colin reeb vector fields and open book decompositions, j. We compute parts of the contact homology of contact 3manifolds which are supported by open books with pseudoanosov monodromy. It is said to be adapted to the dynamics of a vector field x if l consists of periodic orbits, x orients l positively, the pages are global surfaces of section and the orientation of m together with x induce the orientation of the pages. Lusternikschnirelmann theory and closed reeb orbits. V colin, k honda, reeb vector fields and open book decompositions. By previous results it follows that under the above assumptions, there are infinitely many simple reeb orbits if the threemanifold is not the threesphere or a lens space. We also show that for nontorsion contact structures, every nondegenerate contact form has at least four simple reeb orbits. In the case of an open book with pseudoanosov monodromy, suppose the boundary of a page of the open book is connected and the fractional dehn. In section 2 we define open book decompositions of 3manifolds, discuss their ex. The work al2 then implies that all reeb vector fields. Topological entropy for reeb vector fields in dimension three via. Yair minsky yale university asymptotic geometry of the mapping class group.

In r2 c the standard open book decomposition bstd has the origin as binding and the half. Legendrian contact homology and topological entropy journal. Topological entropy for reeb vector fields in dimension three via open book decompositions marcelo r. First, inspired by a construction of colin and honda 9, we show that any open book decomposition with pseudoanosov monodromy, can be. Positive topological entropy of reeb flows on spherizations. Help center detailed answers to any questions you might have. In the case of an open book with pseudoanosov monodromy, suppose the. Pdf reeb vector fields and open book decompositions semantic. We rst discuss some results on open book decompositions proved in 9. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in. Vector fields tangent to a reeb foliation on s3 sciencedirect. Abstractlet m3, g be an almost kenmotsu 3manifold such that the reeb vector field is an eigenvector field of the ricci operator. Given an open book decomposition of a contact three man ifold m.

A contact structure on mis a maximally nonintegrable hyperplane eld tm. On the existence of supporting broken book decompositions for contact forms in dimension 3. Reeb vector fields and open book decompositions 3 were obtained by yau y2, y3. Andras stipsicz, zoltan szabo, the clay mathematics institute and the alfred renyi. Topological entropy for reeb vector fields in dimension three via open book.

This is the starting point of our proof of theorem 1. Topological entropy for reeb vector fields in dimension three via open book decompositions entropie. Topological entropy for reeb vector fields in dimension three via open book decompositions. One needs to introduce a reeb component on a neighborhood of. Research of ko honda ucla department of mathematics.

An open book decomposition of the tight 3sphere with one. This means the reeb field lives in the tangent space to the link, along the positive direction, and r. Meeks, the dynamics theorem for embedded minimal surfaces yair minsky, asymptotic geometry of the mapping class group. Positive topological entropy for reeb flows on 3dimensional. Given an open book decomposition of a contact three manifold m. Jul 12, 2011 on vector fields having properties of reeb fields article pdf available in topological methods in nonlinear analysis 412 july 2011 with 24 reads how we measure reads.

Suppose that there exists a global 1form such that. Topological entropy for reeb vector fields in dimension. This is a companion paper to reeb vector fields and open book decompositions. From one reeb orbit to two cristofarogardiner, daniel and hutchings, michael, journal of. Equivalence of heegaard floer homology and embedded. Finding reeb vector fields associated with contact forms thread starter wwgd. Positive topological entropy of reeb flows on spherizations volume 151 issue 1 leonardo macarini, felix schlenk. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Columbia mathematics department colloquium meets in math 520 every three weeks on wednesday afternoons at 5. Sep 29, 2008 we determine parts of the contact homology of certain contact 3manifolds in the framework of open book decompositions, due to giroux. A conjecture of colin and honda states that the number of periodic reeb orbits of universally tight contact structures on hyperbolic manifolds grows exponentially with the period, and they speculate further that the growth rate of contact homology is polynomial on nonhyperbolic geometries. Some theorems are proved concerning the decomposition of vector fields into gradient and hamiltonian components.

The argument is made using properties of differential forms, in the context of open book decompositions of contact 3manifolds. Algebraic and geometric topology, mathematical sciences publishers. Differential geometry world scientific publishing company. For reeb vector fields on closed 3manifolds, cylindrical contact homology is used to show that the existence of a set of closed reeb orbit. Processes associated with ionic current rectification at a 2dtitanate nanosheet deposit on a microhole poly ethylene terephthalate substrate. What enables us to get a handle on the contact homology is a better understanding of tightness in the open book framework. The seibergwitten equations and the weinstein conjecture ii. As an application we get that if m is a closed irreducible oriented 3manifold that is not a graph manifold, for example a hyperbolic manifold, then every nondegenerate reeb vector field on m has positive topological entropy. On the other hand, we know every contact 3manifold admits a compatible open book, which. 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.

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. We prove that in dimension 3 every nondegenerate contact form is carried by a broken book decomposition. 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. Schlenk positive topological entropy of reeb flows on spherizations, math. Finding reeb vector field associated with a contact form. Essential open book foliations and fractional dehn twist. Frank morgan williams college manifolds with density. Then the nonintegrability condition is equivalent to d n0. Examples are easy to write down on the 3torus and there are highly nontrivial examples of such vector fields on the 3sphere which are counterexamples to the seifert conjecture. On the existence of supporting broken book decompositions for. On the existence of supporting broken book decompositions. Thanks for contributing an answer to mathematics stack exchange. In mathematics, the reeb vector field, named after the french mathematician georges reeb, is a notion that appears in various domains of contact geometry including. We determine parts of the contact homology of certain contact 3manifolds in the framework of open book decompositions, due to giroux.

For an open book with periodic monodromy, we verify the weinstein conjecture. Pdf on vector fields having properties of reeb fields. Transversality of vector fields defined in terms of diff. If a contact manifold arises as a constantenergy hypersurface inside a symplectic manifold, then the reeb vector field is the restriction to the. Extending work of chen, we prove the weinstein conjecture in dimension three for strongly fillable contact structures with either nonvanishing first chern class or with strong and exact filling. Honda reeb vector fields and open book decompositions, j. Our key result is that the spectral invariants are strictly decreasing under the action of the shift operator when periodic orbits are isolated. Jul 11, 2008 among the contributions, readers may find an extensive survey on characteristic classes of riemannian foliations offering also new results, an article showing the uniform simplicity of certain diffeomorphism groups, an exposition of convergences of contact structures to foliations from the point of view of thurstons and thurstonbennequins. Topological entropy for reeb vector fields in dimension three. Mathematics free fulltext on the sign of the curvature. Moreover, we obtain that on a closed 3manifold, every. Here we exhibit a pseudoanosov homeomorphism which acts. For an application see colinhonda, reeb vector fields and open book decompositions.

Weinstein conjecture is about existence of a closed orbit of the reeb vector field on every contact manifold. We show that every open book decomposition of a contact 3manifold can be. Vincent colins 37 research works with 407 citations and 986 reads, including. William meeks university of massachusetts the dynamics theorem for embedded minimal surfaces. Ko honda, reeb vector fields and open book decompositions william h. Project euclid mathematics and statistics online project euclid. Reeb vector fields and open book decompositions core. Zoltan szabo princeton university link floer homology and the. For an open book with periodic monodromy, we verify the weinstein. Reeb vector fields and open book decompositions arxiv. Legendrian contact homology and topological entropy. The contact subbundle is orthogonal to the reeb vector field, and the field of endomorphisms. Transactions of the american mathematical society volume 364. As applications, we quantitatively study the gap between overtwisted contact structures and nonrightveering monodromies.

Finding reeb vector fields associated with contact forms. In this paper we prove that, for large classes of tight contact 3manifolds, hcm. Colloquia are of general mathematical interest and aimed at both faculty and graduate students. As an application, we prove new multiplicity results for simple closed reeb orbits on the standard contact sphere, the unit. It is expected that this conjecture holds for reeb vector fields on contact manifolds of all odd dimensions every reeb vector field has a closed orbit. We are interested in constructing open book decompositions adapted to the dynamics of reeb vector fields. Vincent colins research works university of nantes. On the other hand, we know every contact 3manifold admits a compatible open book, which implies the binding is tangent to the reeb vector field, doesnt this imply the weinstein conjecture. A picture of a simple open book on the tight 3sphere is shown in fig.