Magnetic geodesics, Hodge Laplacian eigenvalues, and isoperimetric inequalities

Cameron Gates Rudd

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arXiv · 2605.03938

Magnetic geodesics, Hodge Laplacian eigenvalues, and isoperimetric inequalities

Cameron Gates Rudd

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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 (and explicit) 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.

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Cameron Gates Rudd. 2026-05-05. Magnetic geodesics, Hodge Laplacian eigenvalues, and isoperimetric inequalities. https://arxiv.org/abs/2605.03938

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