Waist theorems for Tonelli systems in higher dimensions

TL;DR

This paper investigates periodic orbits in higher-dimensional Tonelli Lagrangian systems, establishing the existence of waist orbits and infinitely many periodic orbits near critical energy levels, extending classical geodesic results.

Contribution

It proves the existence of waist orbits on energy levels above the Mañé critical value and shows infinitely many periodic orbits with potential perturbations, generalizing closed geodesic theorems.

Findings

Existence of waist orbits above critical energy levels.

Infinitely many periodic orbits with potential perturbations.

Extension of closed geodesic results to Tonelli systems.

Abstract

We study the periodic orbits problem on energy levels of Tonelli Lagrangian systems over configuration spaces of arbitrary dimension. We show that, when the fundamental group is finite and the Lagrangian has no stationary orbit at the Ma\~n\'e critical energy level, there is a waist on every energy level just above the Ma\~n\'e critical value. With a suitable perturbation with a potential, we show that there are infinitely many periodic orbits on every energy level just above the Ma\~n\'e critical value, and on almost every energy level just below. Finally, we prove the Tonelli analogue of a closed geodesics result due to Ballmann-Thorbergsson-Ziller.