Relative asymptotic behavior of pseudoholomorphic half‐cylinders

By Richard Siefring. 2007

90 citations recorded by the source corpus.

In this article we study the asymptotic behavior of pseudoholomorphic half‐cylinders that converge exponentially to a periodic orbit of a vector field defined by a framed stable Hamiltonian structure. Such maps are of central interest in symplectic field theory and its variants (symplectic Floer homology, contact homology, and embedded contact homology). We prove a precise formula for the asymptotic behavior of the “difference” of two such maps, generalizing results from [6, 7, 12, 15]. Using this result with a technique from [14], we then show that a finite collection of pseudoholomorphic half‐cylinders asymptotic to coverings of a single periodic orbit is smoothly equivalent to solutions to a linear equation. © 2007 Wiley Periodicals, Inc.

Subfields: Biomedical Engineering, Computational Theory and Mathematics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/8fc3b01928cbd16acf0e4d64cf5655cbd24c057d

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Citations in the corpus

A hierarchy of local symplectic filling obstructions for contact $3$-manifolds (2010, background): “in [Hut02] are stated only for a very special class of stable Hamiltonian structures arising from mapping tori, but they extend to the contact case due to the relative asymptotic formulas of Siefring [Sie08]. A HIERARCHY OF FILLING OBSTRUCTIONS FOR CONTACT MANIFOLDS 59 3 1 Figure 18. A planar torsio”
Embedded contact homology and Seiberg–Witten Floer cohomology II (2008, background): “Equation (2-6) can be derived from what is done, for example, in Siefring [19].”
Lectures on Holomorphic Curves in Symplectic and Contact Geometry (2010, unknown): “ two curves as u0(z) = (zq,uˆ0(z)), v0(z) = (zq,ˆv0(z)). 8Our exposition of this topic is heavily inﬂuenced by the asymptotic version of Theorem 2.85, which is a more recent result due to R. Siefring [Sie08] that extends the intersection theory of closed J-holomorphic curves to the punctured case. W”
Beyond ECH capacities (2014, background): “Note that there are only ﬁnitely many such intersections by [22, Corollary 2.5].”
The Restricted Three-Body Problem and Holomorphic Curves (2018, background): “The crucial ingredient which guarantees finiteness of the algebraic intersection number is the following theorem due to Siefring [100].”
On partially wrapped Fukaya categories (2016, background): “In the second, the asymptotics in [27], combined with assumption (4.”
Cylindrical contact homology for dynamically convex contact forms in three dimensions (2014, unknown): “The asymptotic behavior of holomorphic curves described in [12], [25], or [21, Prop.”
Intersection theory of punctured pseudoholomorphic curves (2009, background): “However it is easily seen from the results in [22] and the fact that nondegeneracy implies exponential convergence that if the end is pseudoholomorphic, then there exist positive constants d and M so that |∂̄j,j0ψ(s, t)| = | 1 2 (dψ + j ◦ dψ ◦ j0)(s, t)| ≤ Me −(d+2π)s for all (s, t) ∈ [R,∞)× S1, whe”
Lecture Notes on Embedded Contact Homology (2013, background): “In the present case, parts of the argument require a result of Siefring [51] which generalizes Lemma 5.”
Strongly fillable contact manifolds and $J$-holomorphic foliations (2008, background): “to any nonzero element of kerAx. The union of these eigenspaces for all x∈ P deﬁnes a real line bundle E→ P. The eigenfunctions ofAxappearnaturallyinthe asymptotic formulaproved in [HWZ96b] (see also [Sie08] for a fuller discussion) for a map u∈ M asymptotic to xu∈ P. Choose coordinates (s,t) ∈ [0,∞”