Magnetic ground states and the conformal class of a surface
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866917964557582336 |
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| author | Colbois, Bruno Provenzano, Luigi Savo, Alessandro |
| author_facet | Colbois, Bruno Provenzano, Luigi Savo, Alessandro |
| contents | On a closed, orientable Riemannian surface $Σ_g$ of arbitrary genus $g\geq 1$ and Riemannian metric $h$ we study the magnetic Laplacian with magnetic potential given by a harmonic $1$-form $A$. Its lowest eigenvalue (magnetic ground state energy) is positive, unless $A$ represents an integral cohomology class. We isolate a countable set of ground state energies which we call $\textit{ground state spectrum}$ of the metric $h$. The main result of the paper is to show that the ground state spectrum determines the volume and the conformal class of the metric $h$. In particular, hyperbolic metrics are distinguished by their ground state spectrum.
We also compute the magnetic spectrum of flat tori and introduce some magnetic spectral invariants of $(Σ_g,h)$ which are conformal by definition and involve the geometry of what we call the Jacobian torus of $(Σ_g,h)$ (in Algebraic Geometry, the Jacobian variety of a Riemann surface). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_16940 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Magnetic ground states and the conformal class of a surface Colbois, Bruno Provenzano, Luigi Savo, Alessandro Differential Geometry Spectral Theory 58J50, 58J53, 35P15, 53C18 On a closed, orientable Riemannian surface $Σ_g$ of arbitrary genus $g\geq 1$ and Riemannian metric $h$ we study the magnetic Laplacian with magnetic potential given by a harmonic $1$-form $A$. Its lowest eigenvalue (magnetic ground state energy) is positive, unless $A$ represents an integral cohomology class. We isolate a countable set of ground state energies which we call $\textit{ground state spectrum}$ of the metric $h$. The main result of the paper is to show that the ground state spectrum determines the volume and the conformal class of the metric $h$. In particular, hyperbolic metrics are distinguished by their ground state spectrum. We also compute the magnetic spectrum of flat tori and introduce some magnetic spectral invariants of $(Σ_g,h)$ which are conformal by definition and involve the geometry of what we call the Jacobian torus of $(Σ_g,h)$ (in Algebraic Geometry, the Jacobian variety of a Riemann surface). |
| title | Magnetic ground states and the conformal class of a surface |
| topic | Differential Geometry Spectral Theory 58J50, 58J53, 35P15, 53C18 |
| url | https://arxiv.org/abs/2503.16940 |