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Connected sums and finite energy foliations. I: Contact connected sums. (English) Zbl 1415.53062

Let \(M\) be a closed contact 3-manifold equipped with a nondegenerate contact form \(\lambda\) and a compatible almost complex structure \(J\). A finite energy foliation \(\mathcal{F}\) for the triple \((M,\lambda,J)\) is a set of connected finite-energy pseudo-holomorphic curves with uniformly bounded energies whose images form a smooth foliation of the product \(\mathbb{R}\times M\). The energy \(E(\mathcal{F})\) of the foliation \(\mathcal{F}\) is the supremum of the energies of the curves in this foliation.
The main result is the following:
Theorem. Let \((M,\xi)\) be a contact 3-manifold with contact structure determined by a non-degenerate contact form \(\lambda\), let \(J\in J(M,\lambda)\) be a complex multiplication for which the triple \((M,\lambda,J)\) admits a stable finite energy foliation \(\mathcal{F}\) of energy \(E(\mathcal{F})\). Then there exists an open dense set \(U\subset M\times M\setminus\Delta(M)\) so that for any \((p,q)\in U\) the contact manifold \((M',\xi')\) obtained by performing of a contact connected sum at \((p,q)\) admits a non-degenerate contact from \(\lambda'\) with \(\xi'=\text{ker\,}\lambda'\), a compatible \(J'\) and a stable finite energy foliation \(\mathcal{F}'\) for the triple \((M',\lambda',J')\) with energy \(E(\mathcal{F}')= E(\mathcal{F})\).

MSC:

53D10 Contact manifolds (general theory)
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