Classification of Certain Rational Isoparametric Functions on Damek-Ricci Spaces
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| Format: | Preprint |
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2025
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| _version_ | 1866911020067323904 |
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| author | Csikós, Balázs Horváth, Márton |
| author_facet | Csikós, Balázs Horváth, Márton |
| contents | We classify isoparametric functions on Damek-Ricci spaces which can be written in terms of the standard coordinates $(v,z,t)$ on the half-space model as a polynomial function divided by $t$. Regular level sets of the functions in our classification encompass almost all previously known examples of isoparametric hypersufaces in Damek-Ricci space and also yield new ones. For the new examples, the focal varieties are determined and the mean curvatures of the regular level sets are expressed as a function of the distance from the focal variety. We also study the exceptional case of tubes about $\mathbb R\mathbf H^k$ in $\mathbb C\mathbf H^k$, which are isoparametric, but cannot be obtained as the level sets of any function in our classification. We show that these tubes are level sets of a polynomial function divided by $t^2$, and that analogous functions on Damek-Ricci spaces can be isoparametric only in the case of $\mathbb C\mathbf H^k$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_10608 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Classification of Certain Rational Isoparametric Functions on Damek-Ricci Spaces Csikós, Balázs Horváth, Márton Differential Geometry Primary 53C25, Secondary 53C40, 53C30 We classify isoparametric functions on Damek-Ricci spaces which can be written in terms of the standard coordinates $(v,z,t)$ on the half-space model as a polynomial function divided by $t$. Regular level sets of the functions in our classification encompass almost all previously known examples of isoparametric hypersufaces in Damek-Ricci space and also yield new ones. For the new examples, the focal varieties are determined and the mean curvatures of the regular level sets are expressed as a function of the distance from the focal variety. We also study the exceptional case of tubes about $\mathbb R\mathbf H^k$ in $\mathbb C\mathbf H^k$, which are isoparametric, but cannot be obtained as the level sets of any function in our classification. We show that these tubes are level sets of a polynomial function divided by $t^2$, and that analogous functions on Damek-Ricci spaces can be isoparametric only in the case of $\mathbb C\mathbf H^k$. |
| title | Classification of Certain Rational Isoparametric Functions on Damek-Ricci Spaces |
| topic | Differential Geometry Primary 53C25, Secondary 53C40, 53C30 |
| url | https://arxiv.org/abs/2505.10608 |