Connected sums and finite energy foliations. I: Contact connected sums
type: Item
instance of: scholarly article
title: Connected sums and finite energy foliations. I: Contact connected sums
zbMATH Open document ID: 1415.53062
author: Joel W. Fish
author: Richard Siefring
published in: The Journal of Symplectic Geometry
publication date: 2019-03-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/1311.4221
review text: 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 $E(\mathcal{F}')= E(\mathcal{F})$.
Mathematics Subject Classification ID: 53D10
zbMATH DE Number: 7038269
zbMATH Keywords: finite energy foliation
zbMATH Keywords: almost complex structure
zbMATH Keywords: contact form
zbMATH Keywords: connected sum
zbMATH Keywords: Lagrange point
reviewed by: Mihail B. Banaru
MaRDI profile type: Publication
arXiv ID: 1311.4221
DOI: 10.4310/JSG.2018.V16.N6.A4
description: scientific article