Mañé’s Critical Value in Dynamical Systems

• Mañé’s Critical Value is an energy threshold in Lagrangian and Hamiltonian systems that separates regimes with unbounded action from those with well-defined minimizing measures.
• It plays a crucial role in establishing geodesic completeness, determining the existence of connecting orbits, and supporting quasi-periodic dynamics in weak KAM theory.
• The concept extends to infinite-dimensional settings and informs spectral bounds in quantum systems, thereby bridging variational analysis with geometric and dynamical applications.

Mañé’s Critical Value

Mañé’s critical value is a central concept in the calculus of variations, Hamiltonian dynamics, and weak KAM theory, marking a fundamental energy threshold for the existence of action-minimizing measures, connecting orbits, and the structure of Lagrangian/Hamiltonian systems under various geometric and dynamical regimes. Its different formulations and consequences play a crucial role in finite and infinite-dimensional geometry, spectral theory, and the theory of integrable and partially hyperbolic systems.

1. Variational Definition and Properties

For a Tonelli Lagrangian $L$ on a closed manifold (typically of the form $L(q,v) = \frac{1}{2}|v|^2 - \alpha_q(v)$ for a magnetic system), Mañé’s critical value $c(L)$ is the infimum of energy shifts that render the free-period action functional nonnegative for all closed curves: $$c(L) = \inf \left\{ k \in \mathbb{R}: S_{L+k}(\gamma) \geq 0, \ \forall\ \gamma \text{ closed, any period} \right\}$$ where

$$S_{L+k}(\gamma) = \int_0^T \left[L(\gamma(t), \dot \gamma(t)) + k\right] dt$$

and $\gamma\colon [0, T]\to M$, $\gamma(0) = \gamma(T)$.

In Hamiltonian terms, for $H$ the Fenchel dual of $L$,

$$c(L) = \inf_{u \in C^\infty(M)} \sup_{q \in M} H(q, du(q))$$

This energy threshold separates regimes with radically different dynamical and variational behaviors. For $e < c(L)$, the action functional can become arbitrarily negative; for $e \ge c(L)$, minimizers or minimizing measures exist under the right conditions.

When $L$ has an additional magnetic term, as on a cover $\widehat{M}$ or infinite-dimensional group, the critical value incorporates the magnetic potential and potential energy: $$c(L_{\widehat{M}, \widehat{V}}) = \inf_{f \in C^\infty(\widehat{M})} \sup_{x \in \widehat{M}} \left\{ \frac12 |\widehat{\alpha}(x) + df(x)|^2 + \widehat{V}(x) \right\}$$

In infinite-dimensional group settings with right-invariant metrics and closed two-forms $\sigma$ (magnetic field), the critical value generalizes as

$$c(G, \mathcal{G}, \sigma) = \inf \left\{ \frac12 \|\alpha\|_\infty^2 : d\alpha = \sigma, \ \alpha \text{ (right-invariant primitive)} \right\}$$

2. Geometric and Dynamical Significance

Mañé’s critical value governs the phase transitions of action-minimizing dynamics:

• Hopf–Rinow Phenomenon: On compact or infinite-dimensional manifolds (including half–Lie groups), for energies $k > c(L)$ or $k > c(G, \mathcal{G}, \sigma)$, the geodesic or magnetic geodesic flow is complete, and the analog of the Hopf–Rinow theorem holds: any two points can be connected by a curve minimizing the corresponding action.
• Aubry–Mather and Weak KAM Theory: Above the critical value, the Peierls barrier is finite and the Aubry set (projected minimizing dynamics) is nonempty, typically supporting quasi-periodic orbits and semi-static curves. Below $c(L)$, such variational orbits may disappear, and the action functional is unbounded below.
• Universal Covers: On the universal cover of a manifold, the critical value $c_u(H)$ (for a Hamiltonian $H$) exhibits symplectic invariance and informs the (non-)existence of contact type energy hypersurfaces as well as empty Aubry sets and non-graphical supports for minimizing measures, showing sharper phenomena than in the compact base (Paternain et al., 2013).
• Spectral Consequences: In quantum settings, notably for magnetic Schrödinger operators on covers with amenable deck groups, the ground state energy $\lambda_0$ is bounded above by the Mañé critical value of the classical lifted system: $$\lambda_0(\widehat{M}, \widehat{V}) \leq c(L_{\widehat{M}, \widehat{V}})$$ (Herbrich, 2014)

3. Explicit Computations and Examples

• Odd-dimensional Spheres: For $S^{2n+1}$ with the round metric and standard contact form $\alpha$, the Mañé critical value is explicitly

$c(S^{2n+1}, g, \alpha) = \frac{1}{8}$

Supercritical energies $k > \frac{1}{8}$ ensure that any two points may be joined by a magnetic geodesic (Albers et al., 4 Mar 2025).

• Infinite-dimensional Magnetic Systems: For the magnetic two-component Hunter–Saxton system (M2HS), the critical value is $c(L) = 1/8$ (Maier, 17 Mar 2025). This threshold coincides with the regime where an infinite-dimensional Hopf–Rinow theorem holds, and where global conservative weak solutions can be constructed via completion and embedding into infinite-dimensional spheres.
• Totally Magnetic Submanifolds and Invariants: On spheres, totally magnetic submanifolds are characterized as intersections with complex subspaces; every magnetic geodesic is tangent to a Clifford torus inside a three-sphere (Albers et al., 4 Mar 2025).
• Randers Finsler Metrics on $S^2$: When a criticality assumption is satisfied for Randers metrics, after rescaling the magnetic part the new metric admits infinitely many closed geodesics above the critical value, demonstrating multiplicity of periodic orbits in $(c(L), c_\infty(L))$ (Benedetti et al., 2017).

4. Consequences in Weak KAM and Aubry-Mather Theory

For Mañé Lagrangians on sub-Riemannian tori with Hörmander distributions, the Aubry set, defined using the Peierls barrier

$h(x, y) = \liminf_{T \to \infty} \left[ \text{minimal action over horizontal trajectories from %%%%24%%%% to %%%%25%%%% in time %%%%26%%%%} \right]$

plays a crucial role. If there are finitely many static classes, any viscosity solution (weak KAM solution) is representable as

$$u(x) = \min_{1 \leq i \leq m} \big\{ u(A_i) + h(A_i, x) \big\}$$

A perturbation by horizontal stochastic noise leads, in the small noise limit, to the selection of a unique weak KAM solution corresponding to the invariant (stationary) measure. The Fokker–Planck equation and large deviations establish that $$\lim_{\varepsilon \to 0} -\varepsilon \log m_\varepsilon(x)$$ solves the Hamilton–Jacobi equation associated to the Mañé Lagrangian (Juárez et al., 18 Jan 2024).

5. Higher-level Structural and Genericity Results

• Density and Genericity: Under differentiability assumptions on Mather’s $\beta$-function, the Legendre transforms of rational homology classes become dense in cohomology, providing a “first step” toward Mañé’s conjecture (Massart, 2013). For $C^2$-generic Tonelli Lagrangians on compact surfaces, the Mañé/Aubry sets typically reduce to single hyperbolic periodic orbits, confirming Mañé’s conjecture in this regime (with key roles played by hyperbolicity, Markov partitions, canal perturbations, and structural stability) (Contreras, 2014).
• Symplectic Invariance and New Phenomena: The critical value, Aubry set, and finiteness of the Peierls barrier are invariant under symplectomorphisms of the phase space; nonetheless, on universal covers, classic results like Mather’s graph theorem may fail: the union of minimizing supports may not be a graph (Paternain et al., 2013).

6. Infinite-dimensional Extensions and Geometric Hydrodynamics

• Magnetic Flows on Half–Lie Groups: For right-invariant magnetic flows on half–Lie groups (e.g., diffeomorphism groups with $H^s$ or $C^k$ regularity), the critical value $c(G, \mathcal{G}, \sigma)$ serves as the sharp energy barrier for the Hopf–Rinow property, geodesic completeness, and connectivity by magnetic geodesics. The Finsler metric on the universal cover for $k > c(G, \mathcal{G}, \sigma)$ ensures completeness and existence of the magnetic exponential map (Maier et al., 22 Oct 2025).
• Reduction and Magnetomorphisms: In both finite and infinite dimensions, structures such as the Madelung transform (which becomes a magnetomorphism) and dynamical reduction to totally magnetic three-spheres provide explicit models for the dynamics, blow-up analysis, and global extensions of solutions (Maier, 17 Mar 2025).

7. Table: Mañé’s Critical Value in Major Contexts

Setting	$c(L)$ Formula	Consequence Above $c(L)$
Finite-dim. magnetic	$$\inf_{f \in C^\infty} \sup_x\{\frac12 |\alpha + df|^2 + V(x)\}$$	Hopf–Rinow property, Aubry set nonempty, finite Peierls barrier
Infinite-dim. group	$$\inf \{\frac12\|\alpha\|_\infty^2: d\alpha = \sigma\}$$	Magnetic completeness of flow, geodesic connection, exp map defined
Quantum cover	---	$$\lambda_0(\widehat{M},\widehat{V}) \leq c(L_{\widehat{M},\widehat{V}})$$
$S^{2n+1}$ with std. contact	$1/8$	All points connected by mag. geodesic above $1/8$

8. Broader Impact and Directions

Mañé’s critical value universally marks the dynamical and analytic frontier for geodesic, magnetic, and weak KAM flows in both finite and infinite dimensions. It underpins spectral bounds for quantum systems with magnetic fields (Herbrich, 2014), structural properties of Aubry sets, and phase transitions in variational problems, as well as the integrability and geometric reduction of PDEs and flows on diffeomorphism groups. Current directions involve further generalizations to noncompact and nontrivially covered manifolds, connections to hydrodynamics, and identifying robust mechanisms for the selection and classification of minimizing measures in high-dimensional or infinite-dimensional settings.