We discuss several symplectic aspects related to the Ma\~n\'e critical value
c_u of the universal cover of a Tonelli Hamiltonian. In particular we show that
the critical energy level is never of virtual contact type for manifolds of
dimension greater than or equal to three. We also show the symplectic
invariance of the finiteness of the Peierls barrier and the Aubry set of the
universal cover. We also provide an example where c_u coincides with the
infimum of Mather's \alpha -function but the Aubry set of the universal cover
is empty and the Peierls barrier is finite. A second example exhibits all the
ergodic invariant minimizing measures with zero homotopy, showing that, quite
surprisingly, the union of their supports is not a graph, in contrast with
Mather's celebrated graph theorem.Comment: 25 pages, to appear on NoDE
