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Holomorphic Curves in Symplectic Cobordisms

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Holomorphic Curves in Low Dimensions

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2216))

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Abstract

In this chapter, the focus shifts from closed symplectic 4-manifolds to contact 3-manifolds and symplectic cobordisms between them, starting from a historical overview of the conjectures of Arnold and Weinstein, and leading into the development of Floer homology and symplectic field theory. We then give an overview of useful technical results on punctured holomorphic curves in cobordisms; as in Chap. 2, the focus here is on precise statements rather than proofs, though heuristic explanations are sometimes given.

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Notes

  1. 1.

    The literature is far from unanimous about the necessity of the minus sign in the formula \(\omega (X_{H_t},\cdot ) = -d H_t\), and there are related interdependent sign issues that arise in defining Hamiltonian action functionals, the standard symplectic forms on \({\mathbb R}^{2n}\) and on cotangent bundles, and various other things that depend on these. For further discussion of these issues, see [Wenb] and [MS17, Remark 3.1.6].

  2. 2.

    Note that a symplectically convex hypersurface in \(({\mathbb R}^{2n},\omega _{\operatorname {st}})\) need not be geometrically convex in the usual sense, e.g. every star-shaped hypersurface is symplectically convex.

  3. 3.

    This terminology varies among different authors: some would describe what we are defining here as a “symplectic cobordism from (M +, ξ +) to (M −, ξ −).” This difference of opinion can probably only be resolved on the battlefield.

  4. 4.

    Minor annoyance: the natural orientation of \({\mathbb T}^2 \times {\mathbb R}^2\) is actually the opposite of the one defined by \(\omega _{\operatorname {can}}\) on \(T^*{\mathbb T}^2\). This is the reason for reversing the order of θ and ϕ in (8.10).

  5. 5.

    The word “equivalent” in this context does not mean that Hofer’s definition was the same, but simply that any uniform bound on Hofer’s energy implies a uniform bound on the version defined here, and vice versa. Thus for applications to compactness theory and asymptotics, the two notions are interchangeable.

  6. 6.

    In general, [Hof93] proved that every finite-energy punctured holomorphic curve has positive and negative punctures asymptotic to closed Reeb orbits, but the asymptotic approach to these orbits is much harder to describe if the orbits are not assumed to be at least Morse-Bott. In fact, the asymptotic orbit at each puncture may even fail to be unique up to parametrization, see [Sie17].

  7. 7.

    Requiring \({\mathcal U}\) to have compact closure is useful for technical reasons, as the proof of the theorem requires defining a Banach manifold of perturbed almost complex structures, and there is usually no natural way to put Banach space structures on spaces of maps whose domains are noncompact manifolds.

  8. 8.

    The condition in Proposition 8.50 requiring asymptotic orbits to be distinct and simply covered can be relaxed somewhat, e.g. a version of this result for embedded planes asymptotic to a multiply covered Reeb orbit is used to give a dynamical characterization of the standard contact lens spaces in [HLS15].

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  1. Institut für Mathematik, Humboldt-Universität zu Berlin, Berlin, Germany

    Chris Wendl

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Wendl, C. (2018). Holomorphic Curves in Symplectic Cobordisms. In: Holomorphic Curves in Low Dimensions. Lecture Notes in Mathematics, vol 2216. Springer, Cham. https://doi.org/10.1007/978-3-319-91371-1_8

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