# Flexible and inflexible $CR$ submanifolds

## TL;DR

This paper establishes new embedding results for compactly supported deformations of certain $CR$ submanifolds in complex Euclidean space, showing stability of embeddability under specific pseudoconcavity conditions and providing examples of non-embeddable deformations.

## Contribution

It proves that 2-pseudoconcave $CR$ submanifolds retain embeddability under compact deformations, extending previous quadratic case results and providing counterexamples.

## Key findings

- Stable embeddability for 2-pseudoconcave $CR$ submanifolds
- Extension of previous quadratic $CR$ submanifold results
- Existence of weakly $2$-pseudoconcave $CR$ manifolds with non-embeddable deformations

## 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.