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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On Lewy extension for smooth hypersurfaces in $ \mathbb{C}^n \times\mathbb{R}$


Authors: Jiří Lebl, Alan Noell and Sivaguru Ravisankar
Journal: Trans. Amer. Math. Soc. 371 (2019), 6581-6603
MSC (2010): Primary 32V40; Secondary 32V25
DOI: https://doi.org/10.1090/tran/7605
Published electronically: October 24, 2018
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Abstract: We prove an analogue of the Lewy extension theorem for a real dimension $ 2n$ smooth submanifold $ M \subset \mathbb{C}^{n}\times \mathbb{R}$, $ n \geq 2$. A theorem of Hill and Taiani implies that if $ M$ is CR and the Levi-form has a positive eigenvalue restricted to the leaves of $ \mathbb{C}^n \times \mathbb{R}$, then every smooth CR function $ f$ extends smoothly as a CR function to one side of $ M$. If the Levi-form has eigenvalues of both signs, then $ f$ extends to a neighborhood of $ M$. Our main result concerns CR singular manifolds with a nondegenerate quadratic part $ Q$. A smooth CR $ f$ extends to one side if the Hermitian part of $ Q$ has at least two positive eigenvalues, and $ f$ extends to the other side if the form has at least two negative eigenvalues. We provide examples to show that at least two nonzero eigenvalues in the direction of the extension are needed.


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Additional Information

Jiří Lebl
Affiliation: Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078
Email: lebl@math.okstate.edu

Alan Noell
Affiliation: Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078
Email: noell@math.okstate.edu

Sivaguru Ravisankar
Affiliation: Centre for Applicable Mathematics, Tata Institute of Fundamental Research, Bengaluru 560065, India
Email: sivaguru@tifrbng.res.in

DOI: https://doi.org/10.1090/tran/7605
Keywords: Extension of CR functions, Lewy extension, CR singularity, Levi-flat
Received by editor(s): August 18, 2017
Received by editor(s) in revised form: February 1, 2018
Published electronically: October 24, 2018
Additional Notes: The first author was supported in part by NSF grant DMS-1362337.
Article copyright: © Copyright 2018 American Mathematical Society