Connected sums and finite energy foliations. I: Contact connected sums
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 $E(\mathcal{F}')= E(\mathcal{F})$.
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