Waist theorems for Tonelli systems in higher dimensions (Q785994)

The work is devoted to studying the existence of at least one periodic orbit for Tonelli Lagrangian systems at almost all low energy levels in the case of higher-dimensional configuration spaces. A Tonelli Lagrangian is a smooth function \(L:TM\rightarrow \mathbb{R}\) defined over the tangent bundle of a closed manifold \(M\) (the configuration space) that is fiberwise superlinear with positive-definite fiberwise Hessian. The phase space \(TM\) is laminated into the level sets of the energy function \(E:TM\rightarrow \mathbb{R}\), where by definition, \(E(q,v)=\partial L(q,v)/\partial v-L(q,v)\). The Lagrangian defines a flow \(\varphi_{L}^{t}:TM\rightarrow TM\), that preserves each compact energy level \(E^{-1}(e),e\in \mathbb{R}\). A periodic curve \(\gamma :\mathbb{R}/p\mathbb{Z}\rightarrow M\) lifts to a periodic orbit of the Euler-Lagrange flow on the energy level \(E^{-1}(e)\), if and only if it is a critical point of the corresponding free-period action functional \(S_{e}\), being defined on the space of periodic curves of any possible positive period. Among the critical points of \(S_{e}\), the local minimizers, which are figuratively called \textit{waists}, force important consequences on the Euler-Lagrange dynamics, at least when \(M\) is a surface. The variational properties of the free-period action functional \(S_{e}\) are similar to the ones of the geodesic energy from Finsler geometry when \(e\in \mathbb{R}\) is large, whereas several difficulties arise when \(e\) is low. The authors also show that on every energy level \(e>c_{0}(L)\), the Euler-Lagrange dynamics is conjugated to a Finsler geodesic flow on the unit tangent bundle of \(M\). Then the problem of periodic orbits in this energy range is reduced to the famous closed geodesic problem for closed Finsler manifolds. A waist with energy \(e>c_{0}(L)\) of the Lagrangian system corresponds to a waist of the associated Finsler metric, that is, a closed geodesic that locally minimizes the Finsler length among nearby periodic curves. If the manifold \(M\) is simply connected, a Finsler metric on \(M\) does not necessarily have waists. If \(M\) has non-trivial (and possibly even finite) fundamental group, a Finsler metric on \(M\) always has waists in non-trivial homotopy classes, but it does not necessarily have contractible ones. The related authors' theorem sounds as follows: Theorem. Let \(M\) be a closed manifold with finite fundamental group, and \(L:TM\rightarrow \mathbb{R}\) a Tonelli Lagrangian such that \(e_{0}(L)<c(L)\), where \(c(L)\) is the so-called Mañé critical value. There exists \(c_{w}(L)>c(L)\) such that, for every \(e\in (c(L),c_{w}(L)),L\) possesses at least a contractible waist with energy \(e\). It extends the result of the authors and \textit{G. Benedetti} [``Minimal boundaries in Tonelli Lagrangian systems'', Int. Math. Res. Not. IMRN (to appear) (2017; \url{https://doi.org/10.1093/imrn/rnz246})], except for the simplicity of the waists, to higher-dimensional closed configuration spaces. This allowed them to apply some techniques from 2-dimensional Tonelli dynamics to study the multiplicity of periodic orbits in generic Tonelli Lagrangian systems on arbitrary configuration spaces. On orientable surfaces, the existence of a contractible waist has strong consequences: it often forces the existence of infinitely many other contractible periodic orbits on the same energy level, or on arbitrarily close energy levels. In higher-dimension this phenomenon still exists, but requires stronger assumptions on the waist, for instance the hyperbolicity. Taking into account that the periodic orbit in a given energy level is either hyperbolic or of twist type, the celebrated Birkhoff-Lewis Theorem guarantees that the periodic orbit is an accumulation of periodic orbits on the same energy level. Arguing along this line, the authors obtain the result, stating that there exists a residual subset \(\mathcal{U}\subset C^{\infty}(M)\) such that, for all \(U\in \mathcal{U}\) satisfying \(e_{0}(L-U)<c(L-U)\), the Lagrangian \(L-U\) possesses infinitely many periodic orbits on every energy level \(e\) in an open dense subset of \((c(L-U),c_{w}(L-U))\). In addition, the authors show that when the fundamental group is finite and the Lagrangian has no stationary orbit at the Mañé critical energy level, there is a waist on every energy level just above the Mañé critical value. Based on suitable perturbation with a potential, it is also demonstrated that there are infinitely many periodic orbits on every energy level just above the Mañé critical value, and on almost every energy level just below. Finally, the authors stated the Tonelli analogue of a closed geodesics result due to \textit{W. Ballmann} et al. [Duke Math. J. 48, 585--588 (1981; Zbl 0476.58010)].