Minimal boundaries in Tonelli Lagrangian systems

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 Mañé 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 Mañé 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.

31 pages, 4 figures. This version also incorporates the results of arXiv:1702.08815 (the preprint arXiv:1702.08815 has been withdrawn from the arXiv, and will not be submitted for publication)

References in corpus (2)

• On Tonelli periodic orbits with low energy on surfaces
• Infinitely many periodic orbits just above the Mañé critical value on the 2-sphere

Cited by in corpus (2)

• Fixed energy solutions to the Euler-Lagrange equations of an indefinite Lagrangian with affine Noether charge
• Waist theorems for Tonelli systems in higher dimensions