Fragility and Persistence of Leafwise Intersections

TL;DR

This paper investigates the fragility and persistence of leafwise intersections in coisotropic submanifolds, demonstrating the necessity of contact type conditions and showing that $C^0$-convergence does not guarantee foliation convergence.

Contribution

It constructs examples showing the fragility of leafwise intersections and clarifies the limitations of $C^0$-convergence in symplectic topology.

Findings

Constructed a hypersurface with no leafwise intersections under $C^0$-perturbations.

Showed that contact type conditions are necessary for persistence results.

Demonstrated that $C^0$-convergence of hypersurfaces does not imply convergence of characteristic foliations.

Abstract

In this paper we study the question of fragility and robustness of leafwise intersections of coisotropic submanifolds. Namely, we construct a closed hypersurface and a sequence of Hamiltonians $C^0$-converging to zero such that the hypersurface and its images have no leafwise intersections, showing that some form of the contact type condition on the hypersurface is necessary in several persistence results. In connection with recent results in continuous symplectic topology, we also show that $C^0$-convergence of hypersurfaces, Hamiltonian diffeomorphic to each other, does not in general force $C^0$-convergence of the characteristic foliations.