Coisotropic Intersections

TL;DR

This paper initiates a theoretical framework for coisotropic intersections, demonstrating their persistence under Hamiltonian diffeomorphisms and establishing key properties related to displacement energy and Liouville class.

Contribution

It develops the first theory of coisotropic intersection properties, extending concepts from Lagrangian submanifolds to coisotropic cases under stability conditions.

Findings

Displacement energy of stable coisotropic submanifolds is positive.

Stable, displaceable coisotropic submanifolds have non-zero Liouville class.

Persistence of coisotropic intersections under Hamiltonian diffeomorphisms.

Abstract

In this paper we make the first steps towards developing a theory of intersections of coisotropic submanifolds, similar to that for Lagrangian submanifolds. For coisotropic submanifolds satisfying a certain stability requirement we establish persistence of coisotropic intersections under Hamiltonian diffeomorphisms, akin to the Lagrangian intersection property. To be more specific, we prove that the displacement energy of a stable coisotropic submanifold is positive, provided that the ambient symplectic manifold meets some natural conditions. We also show that a displaceable, stable, coisotropic submanifold has non-zero Liouville class. This result further underlines the analogy between displacement properties of Lagrangian and coisotropic submanifolds.