# Minimal boundaries in Tonelli Lagrangian systems

## TL;DR

This paper establishes the existence of minimal boundaries and simple periodic orbits for Tonelli Lagrangian systems on surfaces, extending Mather's graph theorem and revealing new dynamical properties near critical energy levels.

## Contribution

It introduces the concept of minimal boundaries in this context, extends Mather's graph theorem, and proves the existence of infinitely many closed geodesics for certain Finsler metrics.

## Key findings

- Existence of minimal boundaries for energies between the maximum constant orbit energy and the Mañé critical value.
- Extension of Mather's graph theorem to minimal boundaries in the tangent bundle.
- Existence of infinitely many closed geodesics for specific non-reversible Finsler metrics on the 2-sphere.

## 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.