Magnetic geodesics, Hodge Laplacian eigenvalues, and isoperimetric inequalities

An isoperimetric constant relating length and stable area, or alternatively for hyperbolic manifolds, length and stable commutator length, serves as a Cheeger constant for the smallest eigenvalue of the Hodge Laplacian acting on coexact 1-forms.

Using properties of the magnetic geodesic flow associated to the differential of a coexact eigenform, and its behavior at Mañé's critical energy level, we give new proofs of these Cheeger-like inequalities, with improved (and explicit) constants and volume dependence.

We also make a few observations about the relationship between Mañé's critical values and the eigenvalues, when the manifold is hyperbolic.