The unregularized gradient flow of the symplectic action

@article{Floer1988TheUG,
  title={The unregularized gradient flow of the symplectic action},
  author={Andreas Floer},
  journal={Communications on Pure and Applied Mathematics},
  year={1988},
  volume={41},
  pages={775-813}
}
  • A. Floer
  • Published 1 September 1988
  • Mathematics
  • Communications on Pure and Applied Mathematics
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 this function, we define on a subset of the path space the flow whose trajectories are given by the solutions of the Cauchy-Riemann equation with respect to a suitable almost complex structure on P. In particular, we prove compactness and transversality results for the set of bounded trajectories. 
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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 called
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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 not