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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Levi-flat hypersurfaces with real analytic boundary


Author: Jiří Lebl
Journal: Trans. Amer. Math. Soc. 362 (2010), 6367-6380
MSC (2000): Primary 32V40, 35B65; Secondary 32W20, 32E10, 32D15
Published electronically: July 19, 2010
MathSciNet review: 2678978
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Abstract: Let $ X$ be a Stein manifold of dimension at least 3. Given a compact codimension 2 real analytic submanifold $ M$ of $ X$, that is the boundary of a compact Levi-flat hypersurface $ H$, we study the regularity of $ H$. Suppose that the CR singularities of $ M$ are an $ \mathcal{O}(X)$-convex set. For example, suppose $ M$ has only finitely many CR singularities, which is a generic condition. Then $ H$ must in fact be a real analytic submanifold. If $ M$ is real algebraic, it follows that $ H$ is real algebraic and in fact extends past $ M$, even near CR singularities. To prove these results we provide two variations on a theorem of Malgrange, that a smooth submanifold contained in a real analytic subvariety of the same dimension is itself real analytic. We prove a similar theorem for submanifolds with boundary, and another one for subanalytic sets.


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Additional Information

Jiří Lebl
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
Address at time of publication: Department of Mathematics, University of California at San Diego, La Jolla, California 92093-0112
Email: jlebl@math.uiuc.edu, jlebl@math.ucsd.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-04887-1
PII: S 0002-9947(2010)04887-1
Received by editor(s): November 13, 2007
Received by editor(s) in revised form: July 24, 2008
Published electronically: July 19, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.