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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2506.14033 |
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| _version_ | 1866908586773315584 |
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| author | Herrmann, Hendrik Hsiao, Chin-Yu Lamel, Bernhard |
| author_facet | Herrmann, Hendrik Hsiao, Chin-Yu Lamel, Bernhard |
| contents | Let $(X,T^{1,0}X)$ be a compact strictly pseudoconvex CR manifold which is CR embeddable into the complex Euclidean space. We show that $T^{1,0}X$ can be approximated in $\mathscr{C}^\infty$-topology by a sequence of strictly pseudoconvex CR structures $\{\mathcal{V}^k\}_{k\in \mathbb N}$ such that each $(X,\mathcal{V}^k)$ is CR embeddable into the unit sphere of a complex Euclidean space. Furthermore, as a refinement of this statement, we show that given a one form $α$ on $X$ such that $(X,T^{1,0}X,α)$ is a pseudohermitian manifold we can approximate $(T^{1,0}X,α)$ in $\mathscr{C}^\infty$-topology by a sequence of pseudohermitian structures $\{(\mathcal{V}^k,α^k)\}_{k\in \mathbb N}$ on $X$ such that for each $k\in \mathbb N$ we have that $(X,\mathcal{V}^k,α^k)$ is isomorphic to a real analytic pseudohermitian submanifold of a sphere. A similar result for the Sasakian case was obtained earlier by Loi-Placini. Let $(X,T^{1,0}X,\mathcal{T})$ be a compact Sasakian manifold, i.e. $\mathcal{T}$ is a transversal CR vector field and the one form $α$ defined by $α(\mathcal{T})=1$ and $α(\operatorname{Re}T^{1,0}X)=0$ defines a pseudohermitian structure on $(X,T^{1,0}X)$. Loi-Placini showed that $(T^{1,0}X,\mathcal{T})$ can be smoothly approximated by a sequence of quasi-regular Sasakian structures $\{(\mathcal{V}^k,\mathcal{T}^k)\}_{k\in \mathbb N}$ on $X$ such that each $(X,\mathcal{V}^k,\mathcal{T}^k)$ admits a smooth equivariant CR embedding into a Sasakian sphere. Applying our methods to the Sasakian case we show that it is possible to approximate with a sequence of Sasakian structures having the form $\{(\mathcal{V}^k,\mathcal{T})\}_{k\in \mathbb N}$, i.e. we can keep the vector field $\mathcal{T}$. Further applications concerning Sasakian deformations, the embedding of domains into balls and local approximation results are provided. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_14033 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Approximation of pseudohermitian structures via embeddings into spheres Herrmann, Hendrik Hsiao, Chin-Yu Lamel, Bernhard Complex Variables 32Vxx (Primary) 32A25 (Secondary) Let $(X,T^{1,0}X)$ be a compact strictly pseudoconvex CR manifold which is CR embeddable into the complex Euclidean space. We show that $T^{1,0}X$ can be approximated in $\mathscr{C}^\infty$-topology by a sequence of strictly pseudoconvex CR structures $\{\mathcal{V}^k\}_{k\in \mathbb N}$ such that each $(X,\mathcal{V}^k)$ is CR embeddable into the unit sphere of a complex Euclidean space. Furthermore, as a refinement of this statement, we show that given a one form $α$ on $X$ such that $(X,T^{1,0}X,α)$ is a pseudohermitian manifold we can approximate $(T^{1,0}X,α)$ in $\mathscr{C}^\infty$-topology by a sequence of pseudohermitian structures $\{(\mathcal{V}^k,α^k)\}_{k\in \mathbb N}$ on $X$ such that for each $k\in \mathbb N$ we have that $(X,\mathcal{V}^k,α^k)$ is isomorphic to a real analytic pseudohermitian submanifold of a sphere. A similar result for the Sasakian case was obtained earlier by Loi-Placini. Let $(X,T^{1,0}X,\mathcal{T})$ be a compact Sasakian manifold, i.e. $\mathcal{T}$ is a transversal CR vector field and the one form $α$ defined by $α(\mathcal{T})=1$ and $α(\operatorname{Re}T^{1,0}X)=0$ defines a pseudohermitian structure on $(X,T^{1,0}X)$. Loi-Placini showed that $(T^{1,0}X,\mathcal{T})$ can be smoothly approximated by a sequence of quasi-regular Sasakian structures $\{(\mathcal{V}^k,\mathcal{T}^k)\}_{k\in \mathbb N}$ on $X$ such that each $(X,\mathcal{V}^k,\mathcal{T}^k)$ admits a smooth equivariant CR embedding into a Sasakian sphere. Applying our methods to the Sasakian case we show that it is possible to approximate with a sequence of Sasakian structures having the form $\{(\mathcal{V}^k,\mathcal{T})\}_{k\in \mathbb N}$, i.e. we can keep the vector field $\mathcal{T}$. Further applications concerning Sasakian deformations, the embedding of domains into balls and local approximation results are provided. |
| title | Approximation of pseudohermitian structures via embeddings into spheres |
| topic | Complex Variables 32Vxx (Primary) 32A25 (Secondary) |
| url | https://arxiv.org/abs/2506.14033 |