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Journal Articles International Mathematics Research Notices Year : 2019

Minimal boundaries in Tonelli Lagrangian systems

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Abstract

We prove several new results concerning action minimizing periodic orbits of Tonelli Lagrangian systems on an oriented closed surface $M$. More specifically, we show that for every energy larger than the maximal energy of a constant orbit and smaller than or equal to the Ma\~n\'e critical value of the universal abelian cover, the Lagrangian system admits a minimal boundary, i.e. a global minimizer of the Lagrangian action on the space of smooth boundaries of open sets of $M$. We also extend the celebrated graph theorem of Mather in this context: in the tangent bundle $TM$, the union of the supports of all lifted minimal boundaries with a given energy projects injectively to the base $M$. Finally, we prove the existence of action minimizing simple periodic orbits on energies just above the Ma\~n\'e critical value of the universal abelian cover. This provides in particular a class of non-reversible Finsler metrics on the 2-sphere possessing infinitely many closed geodesics.

Dates and versions

ensl-02352685 , version 1 (07-11-2019)

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Luca A Asselle, Gabriele Benedetti, Marco Mazzucchelli. Minimal boundaries in Tonelli Lagrangian systems. International Mathematics Research Notices, inPress, ⟨10.1093/imrn/rnz246⟩. ⟨ensl-02352685⟩
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