The main theme of this paper is a relative version of the almost
existence theorem for periodic orbits of autonomous Hamiltonian
systems.
¶ We show that almost all low levels of a function on a
geometrically bounded symplectically aspherical manifold carry
contractible periodic orbits of the Hamiltonian flow, provided
that the function attains its minimum along a closed symplectic
submanifold. As an immediate consequence, we obtain the existence
of contractible periodic orbits on almost all low energy levels
for twisted geodesic flows with symplectic magnetic field. We give
examples of functions with a sequence of regular levels without
periodic orbits, converging to an isolated, but very degenerate,
minimum.
¶ The proof of the relative almost existence theorem hinges on the
notion of the relative Hofer-Zehnder capacity and on showing that
this capacity of a small neighborhood of a symplectic submanifold
is finite. The latter is carried out by proving that the flow of a
Hamiltonian with sufficiently large variation has a nontrivial
contractible one-periodic orbit when the Hamiltonian is constant
and equal to its maximum near a symplectic submanifold and
supported in a neighborhood of the submanifold.