| LAGRANGIAN GRAPHS, MINIMIZING MEASURES AND MA N E'S CRITICAL VALUES (2007) | |||||||||||||||
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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 [13]. 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 EulerLagrange flow of L on the energy level k is Anosov, then k must be strictly bigger than the critical value cu(L) of the lift of L to the universal covering of M. It follows that given k! cu(L), there exists a potential / with arbitrarily small C | |||||||||||||||
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