Lagrangian Graphs, Minimizing Measures and Mañé's Critical Values
Lagrangian Graphs, Minimizing Measures and Mañé's Critical Values
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DOI：
10.1007/s000390050074
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发表时间：
1998-11
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影响因子：
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通讯作者：
G. Contreras;R. Iturriaga;G. Paternain;M. Paternain
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文献类型：
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作者：
G. Contreras;R. Iturriaga;G. Paternain;M. Paternain
设是闭连通流形N上的凸超线性拉格朗日量。我们考虑由R. [M3]中的Mañé。我们证明了到覆盖N的提升的临界值等于k值的下确界，使得能量水平k在覆盖的余切丛中限定一个精确的拉格朗日图。因此，我们表明，重新参数化，动态的欧拉-拉格朗日流ofon的能量水平，包含支持最小化措施与非零旋转向量可以减少到芬斯勒度量。我们还证明了，如果在能级上的Euler-Lagrange流是Anosov，则k必须严格大于L提升到N的泛覆盖的临界值λ（）。当k <cu（）时，存在一个具有任意小C2范数的势函数，使得kf的能级具有共轭点。最后，我们证明了N的覆盖的弱KAM解的存在性，并解释了Fathi在[F1，2]中的结果与Mañé的临界值和动作电位之间的关系.
Letbe a convex superlinear Lagrangian on a closed connected manifoldN. We consider critical values of Lagrangians as defined by R. Mañé in [M3]. We show that the critical value of the lift ofto a covering ofNequals the infimum of the values ofksuch that the energy levelkbounds 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 ofon 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 ofon the energy levelkis Anosov, thenkmust be strictly bigger than the critical valuecu() of the lift ofLto the universal covering ofN. It follows that givenk<cu(), there exists a potentialwith arbitrarily smallC2-norm such that the energy levelkofpossesses conjugate points. Finally we show the existence of weak KAM solutions for coverings ofNand we explain the relationship between Fathi's results in [F1,2] and Mañé's critical values and action potentials.