[Paper Review] A remark on a Theorem by Ekeland-Hofer

Peter Albers, Urs Frauenfelder|arXiv (Cornell University)|Jan 19, 2010

Geometric and Algebraic Topology9 citations

TL;DR

This paper extends a result by Ekeland and Hofer by replacing the restricted contact type condition with the weaker star-shaped condition for hypersurfaces in R^{2n}. It proves that under global, centrally symmetric Hamiltonian perturbations, either there are infinitely many leaf-wise intersection points or at least one such point lies on a closed characteristic, thus broadening the applicability of the original theorem.

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.

Motivation & Objective

• To generalize the Ekeland-Hofer theorem from restricted contact type to star-shaped hypersurfaces in R^{2n}.
• To investigate the existence of leaf-wise intersection points under global, centrally symmetric Hamiltonian perturbations.
• To determine whether the number of such points is infinite or if at least one lies on a closed characteristic when the hypersurface is star-shaped.
• To establish a stronger topological result by weakening the geometric assumptions on the hypersurface.

Proposed method

• Replacing the restricted contact type condition with the star-shaped condition in the hypersurface's definition.
• Applying variational methods and critical point theory to analyze leaf-wise intersection points under symmetric perturbations.
• Using the symmetry of the Hamiltonian and hypersurface to exploit invariance properties in the phase space.
• Analyzing the structure of the energy level set to deduce the existence of either infinitely many intersection points or one on a closed characteristic.
• Leveraging the fact that star-shaped hypersurfaces admit a global defining function with positive homogeneity.

Experimental results

Research questions

• RQ1 Does the Ekeland-Hofer theorem on leaf-wise intersection points hold when restricted contact type is replaced by star-shaped hypersurfaces?
• RQ2 Under global, centrally symmetric Hamiltonian perturbations, can we guarantee infinitely many leaf-wise intersection points for star-shaped hypersurfaces?
• RQ3 Is there always at least one leaf-wise intersection point lying on a closed characteristic when the number of such points is finite?
• RQ4 How does the geometric condition of star-shapedness affect the multiplicity and location of leaf-wise intersection points?

Key findings

• The main result generalizes Ekeland-Hofer's theorem by replacing the restricted contact type condition with the weaker star-shaped condition.
• For any global, centrally symmetric Hamiltonian perturbation of a star-shaped hypersurface in R^{2n}, there exist either infinitely many leaf-wise intersection points or at least one such point lies on a closed characteristic.
• The proof relies on symmetry and variational methods to establish the existence of such points without requiring the stronger restricted contact type structure.
• The result shows that the topological obstruction to leaf-wise intersections persists under a broader class of hypersurfaces, enhancing the theorem's scope.