Flexible and inflexible $CR$ submanifolds

Abstract

In this paper we prove new embedding results for compactly supported deformations of $CR$ submanifolds of ${\mathbb{C}}^{n+d}$: We show that if $M$ is a $2$-pseudoconcave $CR$ submanifold of type $(n,d)$ in ${\mathbb{C}}^{n+d}$, then any compactly supported $CR$ deformation stays in the space of globally $CR$ embeddable in ${\mathbb{C}}^{n+d}$ manifolds. This improves an earlier result, where $M$ was assumed to be a quadratic $2$-pseudoconcave $CR$ submanifold of ${\mathbb{C}}^{n+d}$. We also give examples of weakly $2$-pseudoconcave $CR$ manifolds admitting compactly supported $CR$ deformations that are not even locally $CR$ embeddable.

arXiv admin note: text overlap with arXiv:1611.05427