Connected sums and finite energy foliations I: Contact connected sums

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