Arkiv för Matematik

Volume 57 (2019)

Number 1

Flexible and inflexible $CR$ submanifolds

Pages: 23 – 33



Judith Brinkschulte (Mathematisches Institut, Universität Leipzig, Germany)

C. Denson Hill (Department of Mathematics, Stony Brook University, Stony Brook, New York, U.S.A.)


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.


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

2010 Mathematics Subject Classification

32V30, 32V40

Received 16 October 2017

Received revised 1 July 2018

Accepted 14 September 2018

Published 3 May 2019