Inflexible $CR$ submanifolds

TL;DR

This paper introduces the concept of inflexible CR submanifolds, showing that certain 2-pseudoconcave quadratic CR submanifolds are inherently inflexible, meaning their CR structure cannot be deformed into non-embeddable forms.

Contribution

It defines inflexible CR submanifolds and proves that all 2-pseudoconcave quadratic CR submanifolds of a specific type are inflexible.

Findings

2-pseudoconcave quadratic CR submanifolds are inflexible

Inflexibility relates to CR embeddability under deformations

Main theorem applies to submanifolds in complex Euclidean space

Abstract

In this paper we introduce the concept of inflexible $CR$ submanifolds. These are $CR$ submanifolds of some complex Euclidean space such that any compactly supported $CR$ deformation is again globally $CR$ embeddable into some complex Euclidean space. Our main result is that any $2$-pseudoconcave quadratic $CR$ submanifold of type $(n,d)$ in $\mathbb{C}^{n+d}$ is inflexible.