Applications of Hofer's geometry to Hamiltonian dynamics

TL;DR

This paper applies Hofer's geometry to solve problems in Hamiltonian dynamics, proving key conjectures and properties related to contact hypersurfaces, magnetic flows, and Lagrangian submanifolds.

Contribution

It introduces novel applications of Hofer's geometry to establish new results in Hamiltonian dynamics and symplectic topology.

Findings

Weinstein conjecture proven for displaceable contact hypersurfaces.

Existence of contractible closed orbits for magnetic flows at small energies.

Lagrangian submanifolds with certain fundamental groups have the intersection property.

Abstract

We prove the following three results in Hamiltonian dynamics. 1. The Weinstein conjecture holds true for every displaceable hypersurface of contact type. 2. Every magnetic flow on a closed Riemannian manifold has contractible closed orbits for a dense set of small energies. 3. Every closed Lagrangian submanifold of an arbitrary symplectic manifold whose fundamental group injects and which admits a Riemannian metric without closed geodesics has the intersection property.