On boundaries of Levi-flat hypersurfaces in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:msup><mml:mi mathvariant="double-struck">C</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:math>

Dolbeault, Pierre; Tomassini, Giuseppe; Zaitsev, Dmitri

doi:10.1016/j.crma.2005.07.012

Cited by 25 publications

(42 citation statements)

References 9 publications

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“…If n 3, essentially local conditions and the assumption: every complex point of S is elliptic imply the existence of a projection in C n of a Levi-flat (2n − 1)-subvariety whose boundary is S (Dolbeault, Tomassini, Zaitsev, 2005). We extend the result when S is homeomorphic to a sphere and has one hyperbolic point.…”

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confidence: 88%

On Levi-flat hypersurfaces with given boundary in ℂ n

Dolbeault

2008

Sci. China Ser. A-Math.

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Let S ⊂ C n be a compact connected 2-codimensional submanifold. If n 3, essentially local conditions and the assumption: every complex point of S is elliptic imply the existence of a projection in C n of a Levi-flat (2n − 1)-subvariety whose boundary is S (Dolbeault, Tomassini, Zaitsev, 2005). We extend the result when S is homeomorphic to a sphere and has one hyperbolic point.For n = 2 many results are known since the 1980's and a new result with a very technical hypothesis is announced.

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confidence: 88%

On Levi-flat hypersurfaces with given boundary in ℂ n

Dolbeault

2008

Sci. China Ser. A-Math.

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“…The generic (2n − 2)-manifold S is not even locally extendable by a Levi flat hypersurface M . Indeed, locally, S is the graph of a smooth function g, so the existence of a local Levi flat graph extending M amounts to solve a boundary problem for a system of (non-linear) differential operators and this requires compatibility conditions for g. Some existence results in C n have been recently obtained in [18]. Vol.…”

Section: Other Extension Problemsmentioning

confidence: 99%

Recent Results on the Extension Problem of Analytic Objects

Tomassini

2007

Milan j. math.

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We deal with the general problem of extension of analytic objects in a complex space X. After a short presentation of the classical results we discuss some recent developments obtained when X is a semi-1-corona. Semi-1-coronae are domains C + whose boundary is the union of a Levi flat part, a 1-pseudoconvex part and a 1-pseudoconcave part. Using the main result in [31], we prove a "bump lemma" for compact semi-1-coronae in C n and then, applying Andreotti-Grauert theory, we get a cohomology finiteness theorem for coherent sheaves whose depth is at least 3. As an application we get an extension theorem for coherent sheaves and analytic subsets. Mathematics Subject Classification (2000). Primary 32D15; Secondary 32F32, 32T15, 32V15, 32V25, 32W50.

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“…Similarly, the existence result of Dolbeault, Tomassini and Zaitsev [10] does not guarantee a nonsingular hypersurface. An analysis of the proof of Theorem 3.1 allows us to formulate an alternative statement which allows singularities.…”

Section: Hypersurfaces With Singularitiesmentioning

confidence: 99%

“…Our question can be interpreted as a form of a complex Plateau problem; see Bedford [2] for example. The existence question for N ≥ 3 has been considered by Dolbeault, Tomassini and Zaitsev [10].…”

Section: Jiří Leblmentioning

confidence: 99%

Levi-flat hypersurfaces with real analytic boundary

Lebl

2010

Trans. Amer. Math. Soc.

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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 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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On boundaries of Levi-flat hypersurfaces in Cn

Cited by 25 publications

References 9 publications

On Levi-flat hypersurfaces with given boundary in ℂ n

On Levi-flat hypersurfaces with given boundary in ℂ n

Recent Results on the Extension Problem of Analytic Objects

Levi-flat hypersurfaces with real analytic boundary

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