Compactness results in Symplectic Field Theory

By Frederick J. Bourgeois III, Y. Eliashberg, Helmut Hofer, K. Wysocki, E. Zehnder. 2003

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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).

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/90f191d4c402e5df969c5cff873dd439ffaffa1d

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Citations in the corpus

Naturality and Mapping Class Groups in Heegaard Floer Homology (2012, background): “993]) of the curves appearing in the limit, in analogy with the notion of holomorphic twin tower appearing in [6] and [19, Proposition 12.”
Cylindrical contact homology for dynamically convex contact forms in three dimensions (2014, unknown): “Suppose to get a contradiction that there exists a sequence of J - holomorphic cylinders { u ( k ) } in M J 2 ( γ d +1 , γ d ) / R which converges in the sense of [4] to the building ( u 1 , u 2 ).”
Gluing pseudoholomorphic curves along branched covered cylinders II (2007, background): “Note that the definition of #G(U+, U−) does not count boundary points of the compactification of the moduli space M(α+, α−)/R as in [1, 3] (for all we know there could be infinitely many such boundary points), but rather counts boundary points of a truncation of the moduli space.”
Symplectic Floer homology of area-preserving surface diffeomorphisms (2008, background): “5 Directions for further research Cylindrical contact homology Symplectic field theory and contact homology invariants (see Eliashberg, Givental and Hofer [10]) exist for mapping tori due to the existence of a Hamiltonian structure (see Bourgeois et al [3], Cieliebak and Mohnke [5] and Fabert [11]).”
Duality between Lagrangian and Legendrian invariants (2017, methodology): “The theorem then follows from SFT compactness [12].”
Infinitely many monotone Lagrangian tori in del Pezzo surfaces (2016, methodology): “To compute the above invariant, we employed the neck-stretching technique [3,8] to get a degenerated limit of pseudo-holomorphic disks with boundary in T (a2, b2, c2) inside the weighted projective space CP(a2, b2, c2).”
Effect of Legendrian surgery (2009, methodology): “Indeed, Stokes’ theorem provides an upper bound R of the nonnegative energy R C d for any such holomorphic curve C , end hence compactness is guaranteed by a version of Gromov compactness theorem proven by Bourgeois et al [8].”
Lagrangian spheres, symplectic surfaces and the symplectic mapping class group (2010, background): “For detailed expositions on the subject, we refer the readers to [12] and [11].”
Weak and strong fillability of higher dimensional contact manifolds (2011, background): “The point of assuming (λ,Ω) to be a stable Hamiltonian structure is that the compactness results of Symplectic Field Theory [8] are now valid for finite energy J -holomorphic curves in (Ŵ ,ω).”
Rational symplectic field theory over Z2 for exact Lagrangian cobordisms (2006, background): “Details on the subject can be found in [3], [19, 20], and [10, 13].”
Reeb orbits and the minimal discrepancy of an isolated singularity (2014, background): “After passing to a subsequence, one can showvia a compactness argument (see [6] or Proposition 8.”
$S^1$-equivariant symplectic homology and linearized contact homology (2012, background): “To prove the claim, let us note that the relation ∂ 1 ◦ ∂ 1 = 0 for the S(1)equivariant differential ∂ 1 defined on SC 1 ∗ (H , J) can only be violated by the appearance of J-holomorphic buildings with more than one level (in the sense of [7]) on the boundary of 1-dimensional moduli spaces M (γ, γ;H”