Connected sums and finite energy foliations I: Contact connected sums

Abstract

We consider a $3$-manifold $M$ equipped with nondegenerate contact form $\lambda$ and compatible almost complex structure $J$. We show that if the data $(M,\lambda ,J)$ admits a stable finite energy foliation, then for a generic choice of distinct points $p$, $q\in M$, the manifold $M^{\prime }$ formed by taking the connected sum at $p$ and $q$ admits a nondegenerate contact form ${\lambda }^{\prime }$ and compatible almost complex structure $J^{\prime }$ so that the data $(M^{\prime },{\lambda }^{\prime },J^{\prime })$ also admits a stable finite energy foliation. Along the way, we develop some general theory for the study of finite energy foliations.

arXiv comment78 pages, 4 figures; edited to incorporate referee suggestions; accepted for publication in Journal of Symplectic Geometry

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