97-490 Contreras G., Iturriaga R., Paternain G., Paternain M.

Lagrangian graphs, minimizing measures and Mañé's critical values. (33K, AMS-LaTeX) Sep 12, 97

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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 $k<c_{u}(L)$, there exists a potential $\psi$ with arbitrarily small $C^{2}$-norm such that the energy level $k$ of $L+\psi$ possesses conjugate points.

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