Hodge Laplacian on $1$-forms of homogeneous $3$-spheres
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| Format: | Preprint |
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2026
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| _version_ | 1866910195342376960 |
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| author | Henkel, Jonas Lauret, Emilio A. |
| author_facet | Henkel, Jonas Lauret, Emilio A. |
| contents | We study the spectrum of the Hodge-Laplacian on $1$-forms for left-invariant metrics on the Lie group $\operatorname{SU}(2) \cong S^3$ and its quotient $\operatorname{SO}(3)\cong P^3(\mathbb{R})$. To the best of our knowledge, we provide the first explicit computation of the full spectrum of the Hodge-Laplacian for a canonical variation by determining the eigenvalues of Berger 3-spheres and analyzing their resulting splitting behavior. Furthermore, we propose and rigorously prove an explicit formula for the first eigenvalue of general homogeneous metrics on $\operatorname{SU}(2)$ and $\operatorname{SO}(3)$. The formal proof of this result was autonomously discovered by an advanced AI model, providing a notable case study for AI-driven mathematical research. Finally, leveraging this explicit formula, we apply these spectral results to the inverse problem, showing that the spectrum on $1$-forms determines the metric up to isometry. The source code for the symbolic computations, visualizations, and a Monte Carlo stress test is provided in the electronic supplementary material [He26]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_05406 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hodge Laplacian on $1$-forms of homogeneous $3$-spheres Henkel, Jonas Lauret, Emilio A. Differential Geometry Spectral Theory 58C40 (Primary) 58J50, 58J53, 53C30 (Secondary) We study the spectrum of the Hodge-Laplacian on $1$-forms for left-invariant metrics on the Lie group $\operatorname{SU}(2) \cong S^3$ and its quotient $\operatorname{SO}(3)\cong P^3(\mathbb{R})$. To the best of our knowledge, we provide the first explicit computation of the full spectrum of the Hodge-Laplacian for a canonical variation by determining the eigenvalues of Berger 3-spheres and analyzing their resulting splitting behavior. Furthermore, we propose and rigorously prove an explicit formula for the first eigenvalue of general homogeneous metrics on $\operatorname{SU}(2)$ and $\operatorname{SO}(3)$. The formal proof of this result was autonomously discovered by an advanced AI model, providing a notable case study for AI-driven mathematical research. Finally, leveraging this explicit formula, we apply these spectral results to the inverse problem, showing that the spectrum on $1$-forms determines the metric up to isometry. The source code for the symbolic computations, visualizations, and a Monte Carlo stress test is provided in the electronic supplementary material [He26]. |
| title | Hodge Laplacian on $1$-forms of homogeneous $3$-spheres |
| topic | Differential Geometry Spectral Theory 58C40 (Primary) 58J50, 58J53, 53C30 (Secondary) |
| url | https://arxiv.org/abs/2605.05406 |