On the strict Arnold chord property and coisotropic submanifolds of complex projective space

TL;DR

This paper establishes a new intersection property for Legendrian submanifolds under specific conditions and derives bounds on the minimal action of coisotropic submanifolds in complex projective space, impacting embedding problems.

Contribution

It proves a novel intersection result for Legendrian submanifolds with closed characteristics and provides bounds on minimal action in complex projective space, offering new obstructions to embeddings.

Findings

Legendrian submanifolds intersect characteristics at least twice under certain conditions

Minimal action of regular closed coisotropic submanifolds in complex projective space is at most π/2

Obstructions to presymplectic and Lagrangian embeddings are established

Abstract

Let $\alpha$ be a contact form on a manifold $M$, and $L\subseteq M$ a closed Legendrian submanifold. I prove that $L$ intersects some characteristic for $\alpha$ at least twice if all characteristics are closed and of the same period, and $\alpha$ embeds nicely into the product of $\mathbb{R}^{2n}$ and an exact symplectic manifold. As an application of the method of proof, the minimal action of a regular closed coisotropic submanifold of complex projective space is at most $\pi/2$. This yields an obstruction to presymplectic embeddings, and in particular to Lagrangian embeddings.