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. Pdf on vector fields having properties of reeb fields. For an open book with periodic monodromy, we verify the weinstein. Our key result is that the spectral invariants are strictly decreasing under the action of the shift operator when periodic orbits are isolated. Frank morgan williams college manifolds with density. Vector fields tangent to a reeb foliation on s3 sciencedirect. 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. 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 show that every open book decomposition of a contact 3manifold can be. The harmonicity of the reeb vector field with respect to. The contact subbundle is orthogonal to the reeb vector field, and the field of endomorphisms. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Reeb vector fields and open book decompositions core. 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. On the other hand, we know every contact 3manifold admits a compatible open book, which. Equivalence of heegaard floer homology and embedded. 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. We develop a variant of lusternikschnirelmann theory for the shift operator in equivariant floer and symplectic homology. On the existence of supporting broken book decompositions for contact forms in dimension 3.
As an application, we prove new multiplicity results for simple closed reeb orbits on the standard contact sphere, the unit. From one reeb orbit to two cristofarogardiner, daniel and hutchings, michael, journal of. A picture of a simple open book on the tight 3sphere is shown in fig. Positive topological entropy of reeb flows on spherizations. We also show that for nontorsion contact structures, every nondegenerate contact form has at least four simple reeb orbits.
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. First, inspired by a construction of colin and honda 9, we show that any open book decomposition with pseudoanosov monodromy, can be. Stabilizing the monodromy of an open book decomposition. Research of ko honda ucla department of mathematics. Given an open book decomposition of a contact three manifold m. As applications, we quantitatively study the gap between overtwisted contact structures and nonrightveering monodromies. Columbia mathematics department colloquium meets in math 520 every three weeks on wednesday afternoons at 5.
Positive topological entropy for reeb flows on 3dimensional. If a contact manifold arises as a constantenergy hypersurface inside a symplectic manifold, then the reeb vector field is the restriction to the. Positive topological entropy of reeb flows on spherizations volume 151 issue 1 leonardo macarini, felix schlenk. Processes associated with ionic current rectification at a 2dtitanate nanosheet deposit on a microhole poly ethylene terephthalate substrate. Topological entropy for reeb vector fields in dimension three via open book. Equivalence of heegaard floer homology and embedded contact homology via open book decompositions. Here we exhibit a pseudoanosov homeomorphism which acts. 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. Topological entropy for reeb vector fields in dimension three via open book decompositions marcelo r. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in. In this paper we prove that, for large classes of tight contact 3manifolds, hcm. On the existence of supporting broken book decompositions. Differential geometry world scientific publishing company.
Schlenk positive topological entropy of reeb flows on spherizations, math. A constructive method to carry out one of the decompositions is applied to some three. The seibergwitten equations and the weinstein conjecture ii. 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. Then the nonintegrability condition is equivalent to d n0.
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. For an application see colinhonda, reeb vector fields and open book decompositions. Topological entropy for reeb vector fields in dimension three. Vincent colins research works university of nantes. Finding reeb vector field associated with a contact form.
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. Given an open book decomposition of a contact three man ifold m. In the case of an open book with pseudoanosov monodromy, suppose the. Topological entropy for reeb vector fields in dimension. Ko honda, reeb vector fields and open book decompositions william h. For an open book with periodic monodromy, we verify the weinstein conjecture. Thanks for contributing an answer to mathematics stack exchange. Colin reeb vector fields and open book decompositions, j. Planar open book decompositions of 3manifolds onaran, sinem, rocky mountain journal of mathematics, 2014. Suppose that there exists a global 1form such that. William meeks university of massachusetts the dynamics theorem for embedded minimal surfaces.
We are interested in constructing open book decompositions adapted to the dynamics of reeb vector fields. Colloquia are of general mathematical interest and aimed at both faculty and graduate students. Lusternikschnirelmann theory and closed reeb orbits. Legendrian contact homology and topological entropy journal. This means the reeb field lives in the tangent space to the link, along the positive direction, and r. 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.
Honda reeb vector fields and open book decompositions, j. Essential open book foliations and fractional dehn twist. We rst discuss some results on open book decompositions proved in 9. Zoltan szabo princeton university link floer homology and the.
What enables us to get a handle on the contact homology is a better understanding of tightness in the open book framework. Finding reeb vector fields associated with contact forms thread starter wwgd. Pdf reeb vector fields and open book decompositions semantic. We prove that in dimension 3 every nondegenerate contact form is carried by a broken book decomposition. Algebraic and geometric topology, mathematical sciences publishers. 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 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. 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.
Help center detailed answers to any questions you might have. V colin, k honda, reeb vector fields and open book decompositions. 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. Andras stipsicz, zoltan szabo, the clay mathematics institute and the alfred renyi. One needs to introduce a reeb component on a neighborhood of. This is the starting point of our proof of theorem 1.
This is a companion paper to reeb vector fields and open book decompositions. Topological entropy for reeb vector fields in dimension three via open book decompositions. Topological entropy for reeb vector fields in dimension three via open book decompositions entropie. For much more about this see bourgeoisekholmeliashberg, effect of legendrian surgery. For reeb vector fields on closed 3manifolds, cylindrical contact homology is used to show that the existence of a set of closed reeb orbit. A contact structure on mis a maximally nonintegrable hyperplane eld tm. Some theorems are proved concerning the decomposition of vector fields into gradient and hamiltonian components.
We compute parts of the contact homology of contact 3manifolds which are supported by open books with pseudoanosov monodromy. Meeks, the dynamics theorem for embedded minimal surfaces yair minsky, asymptotic geometry of the mapping class group. Open book decompositions and stable hamiltonian structures. Transactions of the american mathematical society volume 364. Project euclid mathematics and statistics online project euclid. In section 2 we define open book decompositions of 3manifolds, discuss their ex. Topological entropy for reeb vector fields in dimension three via. 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. The argument is made using properties of differential forms, in the context of open book decompositions of contact 3manifolds.
The work al2 then implies that all reeb vector fields. We determine parts of the contact homology of certain contact 3manifolds in the framework of open book decompositions, due to giroux. Yair minsky yale university asymptotic geometry of the mapping class group. Transversality of vector fields defined in terms of diff. Reeb vector fields and open book decompositions 3 were obtained by yau y2, y3. On the existence of supporting broken book decompositions for. Abstractlet m3, g be an almost kenmotsu 3manifold such that the reeb vector field is an eigenvector field of the ricci operator. Changing the vector field amounts to isotoping the return map. In r2 c the standard open book decomposition bstd has the origin as binding and the half. S1 is a bration which, when restricted to u, corresponds to the. Legendrian contact homology and topological entropy.
Reeb vector fields and open book decompositions arxiv. Finding reeb vector fields associated with contact forms. Vincent colins 37 research works with 407 citations and 986 reads, including. Moreover, we obtain that on a closed 3manifold, every. Weinstein conjecture is about existence of a closed orbit of the reeb vector field on every contact manifold. Mathematics free fulltext on the sign of the curvature. An open book decomposition of the tight 3sphere with one. Sep 29, 2008 we determine parts of the contact homology of certain contact 3manifolds in the framework of open book decompositions, due to giroux.
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