Cup-length estimates for leaf-wise intersections

Albers, Peter; Momin, Al

doi:10.1017/s0305004110000435

Cited by 15 publications

(21 citation statements)

References 17 publications

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“…So far solutions to Problem 1 have been found under restrictive assumptions on ϕ or N, see [Mo,Ban,EH,Ho2,Gi,Dr,Gü,Zi1,Zi2,AF1,AF2,AF3,AMo,AMc,Bae,Ka1,Ka2,MMP,Sa1,Sa2]. In [Mo, Ban] J. Moser and A. Banyaga assumed that the restriction ϕ| N is C 1 -close to the inclusion N → M. In most other results this condition was relaxed to the condition that ϕ be Hofer close to the identity.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Leafwise Fixed Points for $C^0$-Small Hamiltonian Flows

Ziltener

2017

International Mathematics Research Notices

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Consider a closed coisotropic submanifold N of a symplectic manifold (M, ω) and a Hamiltonian diffeomorphism ϕ on M . The main result of this article states that ϕ has at least the cup-length of N many leafwise fixed points w.r.t. N , provided that it is the time-1-map of a global Hamiltonian flow whose restriction to N stays C 0 -close to the inclusion N → M . If (ϕ, N ) is suitably nondegenerate then the number of these points is bounded below by the sum of the Betti-numbers of N . The nondegeneracy condition is generically satisfied.This appears to be the first leafwise fixed point result in which neither ϕ N is assumed to be C 1 -close to the inclusion N → M , nor N to be of contact type or regular (i.e., "fibering"). It is optimal in the sense that the C 0condition on ϕ cannot be replaced by the assumption that ϕ is Hofer-small.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Leafwise Fixed Points for $C^0$-Small Hamiltonian Flows

Ziltener

2017

International Mathematics Research Notices

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“…Since then the problem of existence of leafwise intersections has been extensively investigated, and Hofer's theorem has been extended to coisotropic submanifolds and to other ambient symplectic manifolds; see, e.g., [AF10,AF12,AMc,AMo,Dr,Gi07,Gü,Ka,Zi09] for an admittedly incomplete but representative list of results on leafwise intersections. A common feature of these results is that, in contrast with Moser's theorem, to ensure the existence of leafwise intersections one has to impose some additional requirements on the hypersurface or the coisotropic submanifold.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Fragility and persistence of leafwise intersections

Ginzburg

Gürel

2015

Math. Z.

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In this paper we study the question of fragility and robustness of leafwise intersections of coisotropic submanifolds. Namely, we construct a closed hypersurface and a sequence of Hamiltonians C 0 -converging to zero such that the hypersurface and its images have no leafwise intersections, showing that some form of the contact type condition on the hypersurface is necessary in several persistence results. In connection with recent results in continuous symplectic topology, we also show that C 0 -convergence of hypersurfaces, Hamiltonian diffeomorphic to each other, does not in general force C 0 -convergence of the characteristic foliations.

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“…The Floer homology RFH of this functional was constructed by Cieliebak and Frauenfelder in [43]. In Theorem 6.3 one leafwise intersection point and assertion (ii) was established in [12], while the cuplength estimate was proven in [14]. RFH was used in [45] to study the dynamics of a charged particle in a magnetic field at different energy levels.…”

Section: The Second One Is the Floer Negative Gradient Equation Whicmentioning

confidence: 99%

Floer Homologies, with Applications

Abbondandolo

Schlenk

2018

Jahresber. Dtsch. Math. Ver.

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Floer invented his theory in the mid eighties in order to prove the Arnol'd conjectures on the number of fixed point of Hamiltonian diffeomorphisms and Lagrangian intersections. Over the last thirty years, many versions of Floer homology have been constructed. In symplectic and contact dynamics and geometry they have become a principal tool, with applications that go far beyond the Arnol'd conjectures: The proof of the Conley conjecture and of many instances of the Weinstein conjecture, rigidity results on Lagrangian submanifolds and on the group of symplectomorphisms, lower bounds for the topological entropy of Reeb flows and obstructions to symplectic embeddings are just some of the applications of Floer's seminal ideas. Other Floer homologies are of topological nature. Among their applications are Property P for knots and the construction of compact topological manifolds of dimension greater than five that are not triangulisable. This is by no means a comprehensive survey on the presently known Floer homologies and their applications. Such a survey would take several hundred pages. We just describe some of the most classical versions and applications, together with the results that we know or like best. The text is written for non-specialists and the focus is on ideas rather than generality. Two intermediate sections recall basic notions and concepts from symplectic dynamics and geometry. 14 4.

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Cup-length estimates for leaf-wise intersections

Abstract: Abstract. We prove that on a restricted contact type hypersurface the number of leaf-wise intersections is bounded from below by a certain cup-length.

Cited by 15 publications

References 17 publications

Leafwise Fixed Points for $C^0$-Small Hamiltonian Flows

Leafwise Fixed Points for $C^0$-Small Hamiltonian Flows

Fragility and persistence of leafwise intersections

Floer Homologies, with Applications

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