In this paper we construct the Floer homology for an action functional which was introduced by Rabinowitz and prove a vanishing theorem. As an application, we show that there are no displaceable exact contact embeddings of the unit cotangent bundle of a sphere of dimension greater than three into a convex exact symplectic manifold with vanishing first Chern class. This generalizes Gromov's result that there are no exact Lagrangian embeddings of a sphere into ރ n .
We present a new method to prove transversality for holomorphic curves in symplectic manifolds, and show how it leads to a definition of genus zero Gromov-Witten invariants. The main idea is to introduce additional marked points that are mapped to a symplectic hypersurface of high degree in order to stabilize the domains of holomorphic maps.
In this paper we define invariants of Hamiltonian group actions for central regular values of the moment map. The key hypotheses are that the moment map is proper and that the ambient manifold is symplectically aspherical. The invariants are based on the symplectic vortex equations. Applications include an existence theorem for relative periodic orbits, a computation for circle actions on a complex vector space, and a theorem about the relation between the invariants introduced here and the Seiberg-Witten invariants of a product of a Riemann surface with a two-sphere.where γ k,d denotes the spin c -structure determined by k and d. Moreover, if k > 2g − 2, then Φ M d,S ,µ d,S k,Σ (c m ) = Φ M k,Σ ,µ k,Σ d,S (c m ). Combining Theorems B and C one can recover the computation of the Seiberg-Witten invariants of product ruled surfaces by Li-Liu [21] and Ohta-Ono [28]. It is also interesting to examine the relation between our invariants and the Gromov-Witten invariants of the symplectic quotient M := M/ /G(τ ) := µ −1 (τ )/G whenever G acts freely on µ −1 (τ ). Such a relation was established in [15] under the hypothesis that the quotient is monotone. Under this condition (and hypotheses (H1 − 3)) it is shown in [15] that there exists a surjective ring homomorphism φ : H * G (M ) → QH * (M ) (with values in the quantum cohomology of the quotient) such that Φ M,µ−τ B,Σ
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.