Defining equations of $7$-dimensional model CR hypersurfaces
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2023
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| _version_ | 1866909567644860416 |
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| author | Gregorovič, Jan Sykes, David |
| author_facet | Gregorovič, Jan Sykes, David |
| contents | We study CR hypersurfaces in $\mathbb{C}^4$ that are Levi degenerate with constant rank Levi form, and moreover finitely nondegenerate. Each of these can be described as a deformation of a model CR hypersurface by adding terms of higher natural weighted order to the model's defining equation. We obtain a complete normal form for models of real analytic uniformly $2$-nondegenerate CR hypersurfaces in $\mathbb{C}^4$, and present a detailed study of their local invariants. The normal form illustrates that $2$-nondegenerate models in $\mathbb{C}^4$ comprise a moduli space parameterized by two univariate holomorphic functions, which is in sharp contrast to the well known Levi-nondegenerate setting and the more recently discovered behavior of $2$-nondegenerate structures in $\mathbb{C}^3$. In further contrast to these previously studied settings, we demonstrate that not all $2$-nondegenerate structures in $\mathbb{C}^4$ arise as perturbations of homogeneous models. We derive defining equations for the homogeneous $2$-nondegenerate models, a set of $9$ structures, and find explicit formulas for their infinitesimal symmetries. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_18588 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Defining equations of $7$-dimensional model CR hypersurfaces Gregorovič, Jan Sykes, David Complex Variables Differential Geometry 32V05, 32V40, 53C30 We study CR hypersurfaces in $\mathbb{C}^4$ that are Levi degenerate with constant rank Levi form, and moreover finitely nondegenerate. Each of these can be described as a deformation of a model CR hypersurface by adding terms of higher natural weighted order to the model's defining equation. We obtain a complete normal form for models of real analytic uniformly $2$-nondegenerate CR hypersurfaces in $\mathbb{C}^4$, and present a detailed study of their local invariants. The normal form illustrates that $2$-nondegenerate models in $\mathbb{C}^4$ comprise a moduli space parameterized by two univariate holomorphic functions, which is in sharp contrast to the well known Levi-nondegenerate setting and the more recently discovered behavior of $2$-nondegenerate structures in $\mathbb{C}^3$. In further contrast to these previously studied settings, we demonstrate that not all $2$-nondegenerate structures in $\mathbb{C}^4$ arise as perturbations of homogeneous models. We derive defining equations for the homogeneous $2$-nondegenerate models, a set of $9$ structures, and find explicit formulas for their infinitesimal symmetries. |
| title | Defining equations of $7$-dimensional model CR hypersurfaces |
| topic | Complex Variables Differential Geometry 32V05, 32V40, 53C30 |
| url | https://arxiv.org/abs/2310.18588 |