Symplectic fixed points and holomorphic spheres

  title={Symplectic fixed points and holomorphic spheres},
  author={Andreas Floer},
  journal={Communications in Mathematical Physics},
  • A. Floer
  • Published 1989
  • Mathematics
  • Communications in Mathematical Physics
LetP be a symplectic manifold whose symplectic form, integrated over the spheres inP, is proportional to its first Chern class. On the loop space ofP, we consider the variational theory of the symplectic action function perturbed by a Hamiltonian term. In particular, we associate to each isolated invariant set of its gradient flow an Abelian group with a cyclic grading. It is shown to have properties similar to the homology of the Conley index in locally compact spaces. As an application, we… Expand
A symplectic fixed point theorem for toric manifolds
In this paper, by a toric manifold we mean a non-singular symplectic quotient M = ℂ n //T k of the standard symplectic space by a linear torus action. Such a toric manifold is in fact a complexExpand
Hamiltonian dynamics on convex symplectic manifolds
We study the dynamics of Hamiltonian diffeomorphisms on convex symplectic manifolds. To this end we first establish an explicit isomorphism between the Floer homology and the Morse homology of such aExpand
Rigid subsets of symplectic manifolds
Abstract We show that there is an hierarchy of intersection rigidity properties of sets in a closed symplectic manifold: some sets cannot be displaced by symplectomorphisms from more sets than theExpand
J-holomorphic curves and symplectic invariants
Given an almost complex structure J on a manifold M, a map f from a Riemann surface to M is called a pseudoholomorphic (or J-holomorphic) curve if at each point p of the surface, the ordinaryExpand
J-holomorphic curves and symplectic invariants
Given an almost complex structure J on a manifold M, a map f from a Riemann surface to M is called a pseudoholomorphic (or J-holomofphic) curve if at each point p of the surface, the ordinaryExpand
Symplectic homology and periodic orbits near symplectic submanifolds
Abstract We show that a small neighborhood of a closed symplectic submanifold in a geometrically bounded aspherical symplectic manifold has non-vanishing symplectic homology. As a consequence, weExpand
Construction of spectral invariants of Hamiltonian diffeomorphisms on general symplectic manifolds
In this paper, we develop a mini-max theory of the action functional over the semi-infinite cycles via the chain level Floer homology theory and construct spectral invariants of HamiltonianExpand
Growth of maps, distortion in groups and symplectic geometry
Abstract.In the present paper we study two sequences of real numbers associated to a symplectic diffeomorphism:¶• The uniform norm of the differential of its n-th iteration;¶• The word length of itsExpand
Positivity of Symplectic Area for Perturbed J-holomorphic Curves
In this paper we will prove that for a compact, symplectic manifold $(M, \omega)$ and for $\omega$-compatible almost-complex structure J any properly perturbed J-holomorphic curve has a non-negativeExpand
Calabi quasimorphism and quantum homology
We prove that the group of area-preserving diffeomorphisms of the 2-sphere admits a non-trivial homogeneous quasimorphism to the real numbers with the following property. Its value on anyExpand


A relative morse index for the symplectic action
The notion of a Morse index of a function on a finite-dimensional manifold cannot be generalized directly to the symplectic action function a on the loop space of a manifold. In this paper, we defineExpand
Reduction of symplectic manifolds with symmetry
We give a unified framework for the construction of symplectic manifolds from systems with symmetries. Several physical and mathematical examples are given; for instance, we obtain Kostant’s resultExpand
The unregularized gradient flow of the symplectic action
The symplectic action can be defined on the space of smooth paths in a symplectic manifold P which join two Lagrangian submanifolds of P. To pursue a new approach to the variational theory of thisExpand
Pseudo holomorphic curves in symplectic manifolds
Definitions. A parametrized (pseudo holomorphic) J-curve in an almost complex manifold (IS, J) is a holomorphic map of a Riemann surface into Is, say f : (S, J3 ~(V, J). The image C=f(S)C V is calledExpand
A symplectic fixed point theorem for ℂℙn
SummaryTwo symplectic diffeomorphisms,φ0,φ1 of a symplectic manifold (X, ω) are said to be homologous if there exists a smooth homotopyφ1,t∋[0, 1] of symplectic diffeomorphisms between them such thatExpand
Lagrangian embeddings and critical point theory
Abstract We derive a lower bound for the number of intersection points of an exact Lagrangian embedding of a compact manifold into its cotangent bundle with the zero section. To do this theExpand
The Birkhoff-Lewis fixed point theorem and a conjecture of V.I. Arnold
Abstract : The following conjecture of V. I. Arnold is proved: every measure preserving diffeomorphism of the torus T2, which is homologeous to the identity, and which leaves the center of massExpand
A refinement of the Conley index and an application to the stability of hyperbolic invariant sets
A compact and isolated invariant set of a continuous flow possesses a so called Conley index, which is the homotopy type of a pointed compact space. For this index a well known continuation propertyExpand
Examples of symplectic structures
SummaryIn this paper we construct symplectic forms $$\tilde \omega _k , k \geqq 0$$ , on a compact manifold $${\tilde Y}$$ which have the same homotopy theoretic invariants, but which are notExpand
A Morse equation in Conley's index theory for semiflows on metric spaces
Given a compact (two-sided) flow, an isolated invariant set S and a Morse-decomposition ( M 1 , …, M n ) of S , there is a generalized Morse equation, proved by Conley and Zehnder, which relates theExpand