Magnetic geodesics, Hodge Laplacian eigenvalues, and isoperimetric inequalities
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918485009891328 |
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| author | Rudd, Cameron Gates |
| author_facet | Rudd, Cameron Gates |
| contents | 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 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_03938 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Magnetic geodesics, Hodge Laplacian eigenvalues, and isoperimetric inequalities Rudd, Cameron Gates Geometric Topology Differential Geometry Dynamical Systems 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 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. |
| title | Magnetic geodesics, Hodge Laplacian eigenvalues, and isoperimetric inequalities |
| topic | Geometric Topology Differential Geometry Dynamical Systems |
| url | https://arxiv.org/abs/2605.03938 |