Approximation of pseudohermitian structures via embeddings into spheres

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