Approximation of pseudohermitian structures via embeddings into spheres

TL;DR

This paper demonstrates that strictly pseudoconvex CR and pseudohermitian structures on compact manifolds can be approximated by structures embeddable into spheres, enabling better understanding and manipulation of their geometric properties.

Contribution

It introduces a method to approximate CR and pseudohermitian structures by embeddable structures into spheres, extending previous results and applying to Sasakian manifolds with preserved vector fields.

Findings

CR structures can be approximated by embeddable structures into spheres.

Pseudohermitian structures can be approximated by real analytic structures in the sphere.

Approximation results apply to Sasakian manifolds, preserving certain vector fields.

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 $\mathscr{C}^\infty$-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.