Approximation of pseudohermitian structures via embeddings into spheres
Hendrik Herrmann, Chin-Yu Hsiao, Bernhard Lamel
TL;DR
The paper addresses the problem of approximating a given strictly pseudoconvex CR structure by ones arising from real-analytic CR submanifolds of spheres. It uses semi-classical Szeg\ő kernel and Toeplitz operator techniques to build sphere-embeddable deformations that converge in $\mathscr{C}^\infty$ to the original pseudohermitian data, providing both global and local results. In the Sasakian case it extends Loi–Placini by fixing the Reeb field and obtaining deformations induced by weighted spheres, while in the pseudohermitian setting it achieves a parallel approximation via domain-boundary analysis and shows a contrast with finite-dimensional sphere embeddings. Local versions ensure that locally embeddable CR manifolds admit sphere-embeddable deformations in neighborhoods, reinforcing the geometric role of sphere embeddings as a universal model. Overall, the work advances deformation theory and embedding theory for CR and Sasakian geometries with concrete analytic constructions and convergence statements.
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 $α$ 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.