On a Theorem by Ekeland-Hofer

TL;DR

This paper extends a theorem by Ekeland-Hofer, showing that for star-shaped hypersurfaces in symplectic space, there are infinitely many leaf-wise intersection points or at least one on a closed characteristic.

Contribution

It generalizes the original theorem from restricted contact type to star-shaped hypersurfaces, establishing the existence of multiple or special leaf-wise intersection points.

Findings

Infinitely many leaf-wise intersection points exist for star-shaped hypersurfaces.

Alternatively, a leaf-wise intersection point can be found on a closed characteristic.

The result broadens the class of hypersurfaces where leaf-wise intersections are guaranteed.

Abstract

In [EH89, Theorem 1] Ekeland-Hofer prove that for a centrally symmetric, restricted contact type hypersurface in R^{2n} and for any global, centrally symmetric Hamiltonian perturbation there exists a leaf-wise intersection point. In this note we show that if we replace restricted contact type by star-shaped there exists infinitely many leaf-wise intersection points or a leaf-wise intersection point on a closed characteristic.