[Paper Review] On the stability of Mañé critical hypersurfaces

Leonardo Macarini, Gabriel P. Paternain|ArXiv.org|Oct 29, 2009

Advanced Differential Equations and Dynamical Systems8 references3 citations

TL;DR

This paper constructs explicit examples of Tonelli Hamiltonians on the torus 𝕋ⁿ (n ≥ 2) for which the Mañé critical hypersurface is stable (geodesible), resolving a long-standing question about the stability of these energy levels. It establishes a criterion for instability based on the absence of closed orbits in certain free homotopy classes, and proves that any stable energy level of a Tonelli Hamiltonian must contain at least one closed orbit.

ABSTRACT

We construct examples of Tonelli Hamiltonians on $\T^n$ (for any $n\geq 2$) such that the hypersurfaces corresponding to the Mañé critical value are stable (i.e. geodesible). We also provide a criterion for instability in terms of closed orbits in free homotopy classes and we show that any stable energy level of a Tonelli Hamiltonian must contain a closed orbit.

Motivation & Objective

• To determine whether Mañé critical hypersurfaces can be stable (geodesible) for Tonelli Hamiltonians on manifolds other than 𝕋².
• To provide a criterion for instability of the critical hypersurface Σ_{c_u} based on the absence of closed orbits in specific free homotopy classes.
• To prove that any stable energy level of a Tonelli Hamiltonian must contain at least one closed orbit.
• To construct explicit examples of Tonelli Hamiltonians on 𝕋ⁿ (n ≥ 2) for which the Mañé critical hypersurface is stable.

Proposed method

• Leverages the structure of left-invariant Hamiltonians on the solvable Lie group Sol to construct explicit examples on the 3-manifold M = Γ\Sol.
• Uses the Hamiltonian H = ½|p + θ_x|² with |θ| = 1 and θ = e^{-z}dx to define a Tonelli Hamiltonian on M.
• Applies the criterion that if no closed orbit projects to a given free homotopy class Γ, then Σ_{c_u} is unstable.
• Employs the fact that c_u(H) = c_0(H) = 1/2 for amenable groups like Sol, ensuring the critical value is well-defined and regular.
• Uses a recursive construction via Lemma 2.1 and Lemma 2.2 to lift stable examples from 𝕋² to higher-dimensional tori 𝕋ⁿ.
• Analyzes the dynamics of the Hamiltonian vector field on Σ_{1/2} to prove that all closed orbits are homologous to zero, implying instability via the instability criterion.

Experimental results

Research questions

• RQ1 Can Mañé critical hypersurfaces be stable for Tonelli Hamiltonians on 𝕋ⁿ when n ≥ 2?
• RQ2 Is there a topological or dynamical obstruction to the stability of Σ_{c_u} in terms of closed orbits in free homotopy classes?
• RQ3 Must every stable energy level of a Tonelli Hamiltonian contain at least one closed orbit?
• RQ4 Can stable critical hypersurfaces be constructed explicitly on higher-dimensional tori?

Key findings

• The paper constructs explicit examples of Tonelli Hamiltonians on 𝕋ⁿ for any n ≥ 2 such that the Mañé critical hypersurface Σ_c is stable (geodesible).
• It proves that if there exists a free homotopy class Γ in M with no closed orbit of energy c_u projecting to Γ, then Σ_{c_u} is unstable.
• It establishes that any stable energy level of a Tonelli Hamiltonian must contain at least one closed orbit.
• For the solvable Lie group Sol with a cocompact lattice, the Hamiltonian H = ½|p + θ_x|² with |θ| = 1 yields c_u(H) = c_0(H) = 1/2 and all closed orbits in Σ_{1/2} are homologous to zero, implying instability.
• The construction generalizes from 𝕋² to 𝕋ⁿ via iterative lifting, yielding stable critical hypersurfaces in all dimensions n ≥ 2.
• The example on M = Γ\Sol shows that Σ_{1/2} is unstable despite being of contact type, due to the absence of nontrivial closed orbits in nontrivial homotopy classes.