Contreras G., Iturriaga R., Paternain G., Paternain M.
Lagrangian graphs, minimizing measures and Mañé's critical values.
(33K, AMS-LaTeX)
ABSTRACT. Let $L$ be a convex superlinear Lagrangian
on a closed connected manifold $M$. We consider critical
values of Lagrangians as defined by R. Ma\~n\'e in \cite{Ma}.
We show that the critical value of the lift of $L$ to a covering of $M$ equals the infimum
of the values of $k$ such that the energy level $k$
bounds an exact Lagrangian graph in the cotangent bundle
of the covering. As a consequence we show that up
to reparametrization, the dynamics of the Euler-Lagrange flow
of $L$ on an energy level that contains
minimizing measures with
nonzero homology can be reduced to Finsler metrics.
We also show that if the Euler-Lagrange flow of $L$ on the energy
level $k$ is Anosov, then $k$ must be strictly bigger than
the critical value $c_{u}(L)$
of the lift of $L$ to the universal covering
of $M$.
It follows that given $k