Flexible and inflexible $CR$ submanifolds

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.

Keywords

inflexible $CR$ submanifolds, deformations of $CR$ manifolds, embeddings of $CR$ manifolds

2010 Mathematics Subject Classification

32V30, 32V40

Full Text (PDF format)

Received 16 October 2017

Received revised 1 July 2018

Accepted 14 September 2018

Published 3 May 2019