# Compactness results in Symplectic Field Theory

@article{Bourgeois2003CompactnessRI, title={Compactness results in Symplectic Field Theory}, author={Fr'ed'eric Bourgeois and Yasha Eliashberg and Helmut Hofer and Kris Wysocki and Eduard Zehnder}, journal={Geometry \& Topology}, year={2003}, volume={7}, pages={799-888} }

This is one in a series of papers devoted to the foundations of Symplectic Field Theory sketched in (4). We prove compactness results for moduli spaces of holomorphic curves arising in Symplectic Field Theory. The theorems generalize Gromov's compactness theorem in (8) as well as compactness theorems in Floer homology theory, (6, 7), and in contact geometry, (9, 19).

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## References

SHOWING 1-10 OF 46 REFERENCES

J-Holomorphic Curves and Quantum Cohomology

- Mathematics
- 1994

Introduction Local behaviour Moduli spaces and transversality Compactness Compactification of moduli spaces Evaluation maps and transversality Gromov-Witten invariants Quantum cohomology Novikov…

The symplectic sum formula for Gromov–Witten invariants

- Mathematics
- 2000

In the symplectic category there is a 'connect sum' operation that glues symplectic manifolds by identifying neighborhoods of embedded codimension two submanifolds. This paper establishes a formula…

Gromov’s Schwarz lemma as an estimate of the gradient for holomorphic curves

- Mathematics
- 1994

In a symplectic manifold (M 2n , ω) with a tamed almost complex structure J, the 2-form ω induces an area form on (immersed) J-curves f: (Σ, i) → (M, J), where (Σ, i) is a Riemann surface which will…

Gromov’s compactness theorem for pseudo holomorphic curves

- Mathematics
- 1994

We give a complete proof for Gromov's compactness theorem for pseudo holomorphic curves both in the case of closed curves and curves with boundary.

An introduction to symplectic topology

- Mathematics
- 1991

Proposition 1.4. (1) Any symplectic vector space has even dimension (2) Any isotropic subspace is contained in a Lagrangian subspace and Lagrangians have dimension equal to half the dimension of the…

The unregularized gradient flow of the symplectic action

- Mathematics
- 1988

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…

A Degeneration Formula of GW-Invariants

- Mathematics
- 2001

This is the second part of the paper "A degeneration of stable morphisms and relative stable morphisms", (math.AG/0009097). In this paper, we constructed the relative Gromov-Witten invariants of a…

Pseudo holomorphic curves in symplectic manifolds

- Mathematics
- 1985

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…

Finite energy foliations of tight three-spheres and Hamiltonian dynamics

- Mathematics
- 2003

Surfaces of sections are a classical tool in the study of 3-dimensional dynamical systems. Their use goes back to the work of Poincare and Birkhoff. In the present paper we give a natural…