Lagrangian Graphs, Minimizing Measures and Ma��'s Critical Values

Abstract

Let L be a convex superlinear Lagrangian on a closed connected manifold N . We consider critical values of Lagrangians as defined by R. Mañé in [M3]. We show that the critical value of the lift of L to a covering of N 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 supports of minimizing measures with non-zero rotation vector 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 N . It follows that given k < c u (L), there exists a potential ψ with arbitrarily small C 2 -norm such that the energy level k of L + ψ possesses conjugate points. Finally we show the existence of weak KAM solutions for coverings of N and we explain the relationship between Fathi's results in [F1,2] and Mañé's critical values and action potentials.

On the Aubry–Mather theory for symbolic dynamics

“…One observes that this formula corresponds in Lagrangian Aubry-Mather theory to the characterization of Mañé's critical value (see theorem A of [8]). Theorem 1 is just a consequence of the Fenchel-Rockafellar theorem.…”

Lectures on the free period Lagrangian action functional

“…The strict Mañé critical value c 0 (L) is not directly related to the geometry of S κ , but has the following important characterization (see [Fat97] and [CIPP98]):…”

Viscosity solutions methods for converse KAM theory

“…Using viscosity solutions (for the general theory of viscosity solution we suggest the references [3,7,15]), one can prove that a weak form of integrability still holds, see, for instance, [2,6,[8][9][10][11][12][13]19], or [20]. As demonstrated in context of homogenization of Hamilton-Jacobi equations, in the classic but unpublished paper by Lions et al [27], equation (1.2) admits viscosity solutions.…”

“…We assume that the Lagrangian has the form 6) in which g ij (x) is a positive definite metric, h i represents the magnetic field and V (x) is the potential energy. This choice of Lagrangians is quite general, as many important examples have the form (1.6).…”

“…Recent results [2,6,[8][9][10][11][12][13], and [19] show that viscosity solutions of (1.2) encode many properties of Mather sets. In particular, if μ is a Mather measure and u solves (…”