On a family of analytic discs attached to a real submanifold $M ⊂ \mathbb{C}^{N+1}$

Burcea, Valentin

doi:10.4310/maa.2013.v20.n1.a4

Cited by 10 publications

(5 citation statements)

References 14 publications

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“…These will be used in §4 and §5 to prove Theorem 3. [Bur2] depend strongly on all the ellipticity of Bishop invariants and requires that the CR orbits in M near the CR singular point form a family of compact strongly pseudoconvex manifolds shrinking down to the complex tangent such that the Harvey-Lawson theorem applies. This is certainly not the case even when one non-elliptic Bishop invariant at the CR singular point appears.…”

Section: Introductionmentioning

confidence: 99%

Flattening of CR singular points and analyticity of the local hull of holomorphy I

Huang

Yin

2015

Math. Ann.

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Section: Introductionmentioning

confidence: 99%

Flattening of CR singular points and analyticity of the local hull of holomorphy I

Huang

Yin

2015

Math. Ann.

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“…The C.-R. Singularities [4], [6], [11], [14], [15], [17], [26] in codimension 2 are important for the area of the analysis of several complex variables. Dolbeault [8], [9], Dolbeault-Tomassini-Zaitsev [10], [11] used the existence of the C.-R. Singularities in order to study the problems of existence and uniqueness of Levi-flat hypersurfaces with prescribed compact boundary [10], [11].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Dolbeault [8], [9], Dolbeault-Tomassini-Zaitsev [10], [11] used the existence of the C.-R. Singularities in order to study the problems of existence and uniqueness of Levi-flat hypersurfaces with prescribed compact boundary [10], [11]. The author [6] constructed a family of analytic discs attached to a class of C.-R. Singular real submanifolds in codimension 2 trying to understand the local hull of holomorphy using methods from Huang-Krantz [15]. Huang-Yin [17], [18] impresivelly exploited the C.-R. structure near the C.-R. singularity [4], [6] in order to study the local hull of holomorphy [17], [18].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The author [6] constructed a family of analytic discs attached to a class of C.-R. Singular real submanifolds in codimension 2 trying to understand the local hull of holomorphy using methods from Huang-Krantz [15]. Huang-Yin [17], [18] impresivelly exploited the C.-R. structure near the C.-R. singularity [4], [6] in order to study the local hull of holomorphy [17], [18]. Gong-Lebl [14] studied C.-R. Singular submanifolds of codimension 2 which are Levi-flat at the C.-R. points.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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CR Singularities and Generalizations of Moser's Theorem I

Burcea¹

2016

Preprint

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“…(3) There is a C k -smooth diffeomorphism j : There has been important work on the complex plateau problem in C n , n ≥ 3, but when S ⊂ C n is a real-codimension two Bishop submanifold with nonminimal CR points. In this setting, S is expected to bound a Levi-flat hypersurface M. Here we refer to the work Dolbeaut-Tomassini-Zaitsev ( [14], [15]) and Lebl-Noell-Ravisankar ( [33]) for the construction of M, and Huang-Yin ( [28], [29]), Valentin Burcea ([8], [9]), and Fang-Huang ( [17]) for the regularity of M at the CR points of S . The problem can also be formulated as a boundary value problem for a certain degenerate elliptic equation (called the Levi equation) and approached from a PDE point of view.…”

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confidence: 99%

Stability of the hull(s) of an $n$-sphere in $\mathbb{C}^n$

Gupta,

Wawrzyniak

2020

Preprint

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We study the (global) Bishop problem for small perturbations of S n -the unit sphere of C × R n−1 -in C n . We show that if S ⊂ C n is a sufficiently-small perturbation of S n (in the C 3 -norm), then S bounds an (n + 1)-dimensional ball M ⊂ C n that is foliated by analytic disks attached to S . Furthermore, if S is either smooth or real analytic, then so is M (upto its boundary). Finally, if S is real analytic (and satisfies a mild condition), then M is both the envelope of holomorphy and the polynomially convex hull of S . This generalizes the previously known case of n = 2 (CR singularities are isolated) to higher dimensions (CR singularities are nonisolated).PURVI GUPTA AND CHLOE URBANSKI WAWRZYNIAK a submanifold S ⊂ C n and a sufficiently small neighborhood E ⊂ S of p, under what conditions will E be a smooth Levi-flat submanifold with E as part of its smooth boundary? This is known as the Bishop problem. The global version of this problem is typically studied for generic closed submanifolds of C n admitting only nondegenerate CR singularities, i.e, points where the maximal complex tangent space has nongeneric dimension. These are the so-called Bishop submanifolds. If S is totally real (at p), then S is holomorphically convex (at p). Bishop discovered that for an n-manifold S in C n , if S at p has a CR singularity of elliptic type, then S is nontrivial due to the presence of embedded complex analytic disks, or Bishop disks, attached to S near p. Further, he conjectured that these disks foliate a unique Levi-flat submanifold that serves as E for a sufficiently small neighborhood E ⊂ S of p, and contains E as part of its smooth (real analytic) boundary when S is smooth (real analytic) near p. In the case of C 2 , Bishop's conjecture was settled by ) in the smooth category, and by ), Moser ([34]) and Huang-Krantz ([27]) in the real analytic category. For general n, Bishop's conjecture was finally shown to be true in the work of Huang ([25]), partially based on the previous work of ). In contrast to the ellipitc case, Forstnerič-Stout showed in [19] that if p is a hyperbolic complex point of a surface S ⊂ C 2 , then the local envelope of holomorphy of S at p is trivial.The first major breakthrough for the global version of the Bishop problem was made by Bedford and Gaveau, when they proved in [4] that any smooth real Bishop surface S ⊂ C 2 with only two elliptic complex tangent points (hence, S is a sphere) and contained in the boundary of a bounded strongly pseudoconvex domain bounds a Levi-flat hypersurface M that is foliated by embedded analytic disks attached to S . Moreover M is precisely S (and even S , in some cases). By the aforementioned work of Kenig-Webster, Moser and Huang-Moser, S is smooth (real analytic) up to S when S is smooth (real analytic). Subsequently, this result was generalized by Bedford-Klingenberg in [5] and Kruzhilin in [32], who showed that S is a Levi-flat hypersurface bound by S , even when S is a sphere admitting hyperbolic complex tangent points (of course, one cannot expect ful...

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On a family of analytic discs attached to a real submanifold $M ⊂ \mathbb{C}^{N+1}$

Cited by 10 publications

References 14 publications

Flattening of CR singular points and analyticity of the local hull of holomorphy I

Flattening of CR singular points and analyticity of the local hull of holomorphy I

CR Singularities and Generalizations of Moser's Theorem I

Stability of the hull(s) of an $n$-sphere in $\mathbb{C}^n$

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