Minimal Boundaries in Tonelli Lagrangian Systems

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 Mañe critical value of the universal abelian cover, the Lagrangian system admits a minimal boundary, that is, 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 $\textrm{T} M$, 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 Mañe critical value of the universal abelian cover. This provides in particular a class of nonreversible Finsler metrics on the two-sphere possessing infinitely many closed geodesics.