On CR embeddings of strictly pseudoconvex hypersurfaces into spheres in low dimensions
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- by Peter Ebenfelt and André Minor PDF
- Trans. Amer. Math. Soc. 366 (2014), 5693-5706 Request permission
Abstract:
It follows from the 2004 work of the first author, X.Huang, and D. Zaitsev that any local CR embedding $f$ of a strictly pseudoconvex hypersurface $M^{2n+1}\subset \mathbb {C}^{n+1}$ into the sphere $\mathbb {S}^{2N+1}\subset \mathbb {C}^{N+1}$ is rigid, i.e. any other such local embedding is obtained from $f$ by composition with an automorphism of the target sphere $\mathbb {S}^{2N+1}$, provided that the codimension $N-n<n/2$. In this paper, we consider the limit case $N-n=n/2$ in the simplest situation where $n=2$, i.e. we consider local CR embeddings $f\colon M^5\to \mathbb {S}^7$. We show that there are at most two different local embeddings up to composition with an automorphism of $\mathbb {S}^7$. We also identify a subclass of 5-dimensional, strictly pseudoconvex hypersurfaces $M^5$ in terms of their CR curvatures such that rigidity holds for local CR embeddings $f\colon M^5\to \mathbb {S}^7$.References
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Additional Information
- Peter Ebenfelt
- Affiliation: Department of Mathematics, University of California at San Diego, La Jolla, California 92093-0112
- MR Author ID: 339422
- Email: pebenfel@math.ucsd.edu
- André Minor
- Affiliation: Department of Mathematics, University of California at San Diego, La Jolla, California 92093-0112
- Email: aminor@math.ucsd.edu
- Received by editor(s): August 17, 2012
- Published electronically: May 21, 2014
- Additional Notes: The authors were supported in part by DMS-1001322.
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 366 (2014), 5693-5706
- MSC (2010): Primary 32H02, 32V30
- DOI: https://doi.org/10.1090/S0002-9947-2014-06085-6
- MathSciNet review: 3256180