$J$-holomorphic disks with pre-Lagrangian boundary conditions

Abstract

The purpose of this paper is to carry out a classical construction of a non-constant holomorphic disk with boundary on (the suspension of) a Lagrangian submanifold in $\mathbb{R}^{2 n}$ in the case the Lagrangian is the lift of a coisotropic (a.k.a. pre-Lagrangian) submanifold in (a subset $U$ of) $\mathbb{R}^{2 n - 1}$. We show that the positive lower and finite upper bounds for the area of such a disk (which are due to M. Gromov and J.-C. Sikorav and F. Laudenbach-Sikorav for general Lagrangians) depend on the coisotropic submanifold only but not on its lift to the symplectization. The main application is to a $C^0$-characterization of contact embeddings in terms of coisotropic embeddings in another paper by the present author. Moreover, we prove a version of Gromov's non-existence of exact Lagrangian embeddings into standard $\mathbb{R}^{2 n}$ for coisotropic embeddings into $S^1 \times \mathbb{R}^{2 n}$. This allows us to distinguish different contact structures on the latter by means of the (modified) contact shape invariant. As in the general Lagrangian case, all of the existence results are based on Gromov's theory of $J$-holomorphic curves and his compactness theorem (or persistence principle). Analytical difficulties arise mainly at the ends of the cone $\mathbb{R}_+ \times U$.