Magnetic geodesics, Hodge Laplacian eigenvalues, and isoperimetric inequalities

TL;DR

This paper explores inequalities relating geometric and spectral properties of hyperbolic manifolds using magnetic geodesic flow, providing new proofs with improved constants and volume dependence.

Contribution

It introduces novel proofs of Cheeger-like inequalities for the Hodge Laplacian on hyperbolic manifolds utilizing magnetic geodesic flow properties.

Findings

Established new bounds for the smallest eigenvalue of the Hodge Laplacian.

Connected Mañé's critical energy levels with eigenvalues in hyperbolic manifolds.

Provided improved constants and volume dependence in 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.