Magnetic geodesics, Hodge Laplacian eigenvalues, and isoperimetric inequalities

Abstract

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 Ma\~n\'e's critical energy level, we give new proofs of these Cheeger-like inequalities, with improved constants and volume dependence. We also make a few observations about the relationship between Ma\~n\'e's critical values and the eigenvalues, when the manifold is hyperbolic.