Connected sums and finite energy foliations I: Contact connected sums

We consider a $3$-manifold $M$ equipped with a 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 contact 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.

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J. W. Fish’s research was partially supported by NSF grant DMS-0844188 and the Ellentuck Fund. Richard Siefring’s research was partially supported by DFG grant BR 5251/1-1.

Received 6 February 2014

Accepted 23 February 2018

Published 18 March 2019