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

TL;DR

This paper constructs specific $J$-holomorphic disks with boundary conditions related to coisotropic submanifolds, revealing bounds on their area and applications to contact embedding characterizations and contact structure distinctions.

Contribution

It provides a classical construction of holomorphic disks with pre-Lagrangian boundary conditions and analyzes their area bounds, extending Gromov's theory to coisotropic submanifolds.

Findings

Area bounds depend only on the coisotropic submanifold, not its lift.

Constructs holomorphic disks with boundary on lifted coisotropic submanifolds.

Distinguishes contact structures via the contact shape invariant.

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 $\mathbb{R}_+ \times U$.