scholarly journals Floer Homology and Gromov-Witten Invariant over Integer of General Symplectic Manifolds – Summary –

2018 ◽  
Author(s):  
Kenji Fukaya ◽  
Kaoru Ono
Keyword(s):  
Floer Homology ◽  
Symplectic Manifolds ◽  
Witten Invariant

scholarly journals Minimal Maslov number of R-spaces canonically embedded in Einstein-Kähler C-spaces

2019 ◽  
Vol 6 (1) ◽  
pp. 303-319
Author(s):  
Yoshihiro Ohnita
Keyword(s):  
Symmetric Space ◽  
Floer Homology ◽  
Lagrangian Submanifolds ◽  
Symplectic Manifolds ◽  
Lagrangian Submanifold ◽  
Riemannian Symmetric Space ◽  
Canonical Embedding ◽  
Real Form ◽  
Isotropy Representation ◽  
Totally Geodesic

AbstractAn R-space is a compact homogeneous space obtained as an orbit of the isotropy representation of a Riemannian symmetric space. It is known that each R-space has the canonical embedding into a Kähler C-space as a real form, and thus a compact embedded totally geodesic Lagrangian submanifold. The minimal Maslov number of Lagrangian submanifolds in symplectic manifolds is one of invariants under Hamiltonian isotopies and very fundamental to study the Floer homology for intersections of Lagrangian submanifolds. In this paper we show a Lie theoretic formula for the minimal Maslov number of R-spaces canonically embedded in Einstein-Kähler C-spaces, and provide some examples of the calculation by the formula.

scholarly journals FLOER HOMOLOGY FOR SYMPLECTOMORPHISM

2009 ◽  
Vol 11 (06) ◽  
pp. 895-936 ◽  
Author(s):  
HAI-LONG HER
Keyword(s):  
Fixed Points ◽  
Symplectic Manifold ◽  
Floer Homology ◽  
Symplectic Manifolds ◽  
Homology Groups ◽  
Symplectic Diffeomorphism ◽  
Novikov Ring ◽  
Compact Symplectic Manifold

Let (M,ω) be a compact symplectic manifold, and ϕ be a symplectic diffeomorphism on M, we define a Floer-type homology FH*(ϕ) which is a generalization of Floer homology for symplectic fixed points defined by Dostoglou and Salamon for monotone symplectic manifolds. These homology groups are modules over a suitable Novikov ring and depend only on ϕ up to a Hamiltonian isotopy.

scholarly journals LEAF-WISE INTERSECTIONS AND RABINOWITZ FLOER HOMOLOGY

2010 ◽  
Vol 02 (01) ◽  
pp. 77-98 ◽  
Author(s):  
PETER ALBERS ◽  
URS FRAUENFELDER
Keyword(s):  
Critical Points ◽  
Floer Homology ◽  
Symplectic Manifolds ◽  
Action Functional ◽  
Multiplicity Results ◽  
Contact Type ◽  
Existence And Multiplicity ◽  
Rabinowitz Floer Homology ◽  
Intersection Points ◽  
Rabinowitz Action Functional

In this paper we explain how critical points of a particular perturbation of the Rabinowitz action functional give rise to leaf-wise intersection points in hypersurfaces of restricted contact type. This is used to derive existence and multiplicity results for leaf-wise intersection points in hypersurfaces of restricted contact type in general exact symplectic manifolds. The notion of leaf-wise intersection points was introduced by Moser [16].

Gromov–Witten type invariants on circle bundles over symplectic manifolds

2016 ◽  
Vol 13 (09) ◽  
pp. 1650114
Author(s):  
Yong Seung Cho ◽  
Young Do Chai
Keyword(s):  
Moduli Space ◽  
Contact Structure ◽  
Quantum Cohomology ◽  
Symplectic Manifolds ◽  
Base Space ◽  
Total Space ◽  
Witten Invariant ◽  
Circle Bundles ◽  
Type Invariant ◽  
The One

We consider circle bundles over symplectic manifolds to study Gromov–Witten type invariants. We investigate the moduli space of pseudo-coholomorphic maps, Gromov–Witten type invariant, the quantum type cohomology of the total space which has a natural contact structure. We then compare Gromov–Witten invariant, and quantum cohomology of the base space with the one of the total space, and derive some relations between them.

scholarly journals Bounds for tentacular Hamiltonians

2018 ◽  
Vol 12 (01) ◽  
pp. 209-265 ◽  
Author(s):  
Federica Pasquotto ◽  
Jagna Wiśniewska
Keyword(s):  
Floer Homology ◽  
Energy Levels ◽  
Symplectic Manifolds ◽  
Rabinowitz Floer Homology ◽  
The Subject ◽  
Definition Of

This paper represents a first step toward the extension of the definition of Rabinowitz Floer homology to non-compact energy hypersurfaces in exact symplectic manifolds. More concretely, we study under which conditions it is possible to establish [Formula: see text]-bounds for the Floer trajectories of a Hamiltonian with non-compact energy levels. Moreover, we introduce a class of Hamiltonians, called tentacular Hamiltonians which satisfy the conditions: how to define Rabinowitz Floer homology for these examples will be the subject of a follow-up paper.

scholarly journals Non-contractible periodic orbits in Hamiltonian dynamics on closed symplectic manifolds

2016 ◽  
Vol 152 (9) ◽  
pp. 1777-1799 ◽  
Author(s):  
Viktor L. Ginzburg ◽  
Başak Z. Gürel
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Symplectic Manifold ◽  
Floer Homology ◽  
Hamiltonian Dynamics ◽  
Symplectic Manifolds ◽  
Homotopy Classes ◽  
Hamiltonian Diffeomorphisms ◽  
Favorable Conditions ◽  
Hamiltonian Diffeomorphism

We study Hamiltonian diffeomorphisms of closed symplectic manifolds with non-contractible periodic orbits. In a variety of settings, we show that the presence of one non-contractible periodic orbit of a Hamiltonian diffeomorphism of a closed toroidally monotone or toroidally negative monotone symplectic manifold implies the existence of infinitely many non-contractible periodic orbits in a specific collection of free homotopy classes. The main new ingredient in the proofs of these results is a filtration of Floer homology by the so-called augmented action. This action is independent of capping and, under favorable conditions, the augmented action filtration for toroidally (negative) monotone manifolds can play the same role as the ordinary action filtration for atoroidal manifolds.

scholarly journals Spectral numbers in Floer theories

2008 ◽  
Vol 144 (6) ◽  
pp. 1581-1592 ◽  
Author(s):  
Michael Usher
Keyword(s):  
Type Theory ◽  
Floer Homology ◽  
Homology Class ◽  
Symplectic Manifolds ◽  
Best Approximations ◽  
Class A ◽  
Chain Complexes ◽  
Definition Of ◽  
Formal Properties ◽  
Free Modules

AbstractThe chain complexes underlying Floer homology theories typically carry a real-valued filtration, allowing one to associate to each Floer homology class a spectral number defined as the infimum of the filtration levels of chains representing that class. These spectral numbers have been studied extensively in the case of Hamiltonian Floer homology by Oh, Schwarz and others. We prove that the spectral number associated to any nonzero Floer homology class is always finite, and that the infimum in the definition of the spectral number is always attained. In the Hamiltonian case, this implies that what is known as the ‘nondegenerate spectrality’ axiom holds on all closed symplectic manifolds. Our proofs are entirely algebraic and apply to any Floer-type theory (including Novikov homology) satisfying certain standard formal properties. The key ingredient is a theorem about the existence of best approximations of arbitrary elements of finitely generated free modules over Novikov rings by elements of prescribed submodules with respect to a certain family of non-Archimedean metrics.

scholarly journals Spectral invariants of distance functions

2016 ◽  
Vol 08 (04) ◽  
pp. 655-676 ◽  
Author(s):  
Suguru Ishikawa
Keyword(s):  
Symplectic Manifold ◽  
Symplectic Form ◽  
Floer Homology ◽  
Symplectic Manifolds ◽  
Distance Functions ◽  
Spectral Invariants ◽  
Euclidian Space ◽  
Closed Riemann Surface ◽  
Genus Formula ◽  
Spectral Invariant

Calculating the spectral invariant of Floer homology of the distance function, we can find new superheavy subsets in symplectic manifolds. We show if convex open subsets in Euclidian space with the standard symplectic form are disjointly embedded in a spherically negative monotone closed symplectic manifold, their complement is superheavy. In particular, the [Formula: see text] bouquet in a closed Riemann surface with genus [Formula: see text] is superheavy. We also prove some analogous properties of a monotone closed symplectic manifold. These can be used to extend Seyfaddni’s result about lower bounds of Poisson bracket invariant.

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