Approximation of pseudohermitian structures via embeddings into spheres

Abstract

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 ${\mathcal{C}}^{\infty }$-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 $\hat{I}\pm$ on $X$ such that $(X,T^{1,0}X,\hat{I}\pm )$ is a pseudohermitian manifold we can approximate $(T^{1,0}X,\hat{I}\pm )$ in ${\mathcal{C}}^{\infty }$-topology by a sequence of pseudohermitian structures $\{({\mathcal{V}}^k,\hat{I}{\pm }^k){\}}_{k\in \mathbb{N}}$ on $X$ such that for each $k\in \mathbb{N}$ we have that $(X,{\mathcal{V}}^k,\hat{I}{\pm }^k)$ 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 $\hat{I}\pm$ defined by $\hat{I}\pm (\mathcal{T})=1$ and $\hat{I}\pm (\mathrm{Re}{T}^{1,0}X)=0$ 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.

65 pages, references added