Abstract
We start this chapter with the simplest SFT compactness result generalizing Gromov compactness. We consider punctured holomorphic curves without boundary in the symplectization of a contact manifold. We define holomorphic buildings and prove the corresponding compactness result with great attention to detail. We then introduce holomorphic buildings for curves with boundary and provide a compactness result. The cases of manifolds with cylindrical ends and symplectic manifolds obtained by splitting along a contact type hypersurface conclude the presentation.
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- 1.
Recall that by definition each \((\tilde{u}_{m},j_{m},\mathbf {S}_{m})\) is itself a sequence \((\tilde{u}^{n}_{m},j^{n}_{m},\mathbf {S}^{n}_{m})_{1\le n\le N_{m}}\), N m ≤N of holomorphic buildings of height 1.
- 2.
The circles and arcs used to compactify the surfaces are removed here.
- 3.
- 4.
This is case (ii) in the paper [12].
- 5.
This is not a real assumption since it can be achieved by merely passing to a suitable subsequence due to the uniform bound on the energy and the nondegeneracy of the contact form. In the paper [12] the authors also consider the Morse–Bott case for curves without boundary. Then one has to assume that all the curves \(\tilde {u}_{n}\) are asymptotic at the corresponding punctures to periodic orbits lying in the same connected component in the space of periodic orbits.
- 6.
Here \(D_{r}=\{z\in{\mathbb{C}}:|z|<r\}\) and \(D^{+}_{r}=\{z\in D_{r}\,|\operatorname{Im}(z)\ge0\}\).
- 7.
The points z ±=(s ±,t ±)∈∂B r (z n ) with \(t_{\pm}=t_{n}\pm\frac{r s_{n}}{|z_{n}|}\) and \(|z_{\pm}|=\sqrt{|z_{n}|^{2}+r^{2}}\) are characterized by the condition that the lines \(\{\tau z_{\pm}\,|\,\tau\in{\mathbb{R}}\}\) are tangent to the boundary of B r (z n ).
- 8.
Recall from the definition of convergence of surfaces that there are diffeomorphisms φ n :S→S n and disjoint embedded loops \(\{\varGamma_{j}\}\subset\dot{S}\) such that φ n (Γ j ) are geodesics in S n . The lengths of these geodesics tend to zero as n→∞. It follows from the proof of the collar lemma that the circles Γ j are actually degenerate boundary components for the limit metric \(h=\lim_{n\rightarrow \infty}\varphi_{n}^{\ast}h_{n}\), and these are isometric to standard cusps. This also explains why removal of singularities theorem and Proposition 2.47 can be applied near a node.
- 9.
Note that a simple calculation yields
$$\varepsilon\ge\min\bigl\{ \rho_n(z)\,|\,z\in{\mathcal{C}}( \gamma_n)\bigr\} =\frac{\ell_n}{2}. $$ - 10.
Also note that S(γ −,γ +,[F])≥0 if the homotopy class [F] has a pseudoholomorphic representative.
- 11.
Recall that by definition each \((\tilde{u}_{m},j_{m},\mathbf {S}_{m})\) is itself a sequence \((\tilde{u}^{n}_{m},j^{n}_{m},\mathbf {S}^{n}_{m})_{1\le n\le N_{m}}\), N m ≤N of holomorphic buildings of height 1.
- 12.
The circles and arcs used to compactify the surfaces are removed here.
- 13.
Note that the ends may be disconnected.
- 14.
This is not a real assumption since it can be achieved by merely passing to a suitable subsequence due to the uniform bound on the energy and the nondegeneracy of the contact form. In the paper [12] the authors also consider the Morse–Bott case for curves without boundary. Then one has to assume that all the curves \(\tilde {u}_{n}\) are asymptotic at the corresponding punctures to periodic orbits lying in the same connected component in the space of periodic orbits.
References
F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki, E. Zehnder, Compactness results in symplectic field theory. Geom. Topol. 7, 799–888 (2003)
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Abbas, C. (2014). The SFT Compactness Results. In: An Introduction to Compactness Results in Symplectic Field Theory. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31543-5_3
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DOI: https://doi.org/10.1007/978-3-642-31543-5_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-31542-8
Online ISBN: 978-3-642-31543-5
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