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

Huang, Xiaojun; Yin, Wanke

doi:10.1007/s00208-015-1228-6

Cited by 22 publications

(19 citation statements)

References 26 publications

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“…The problem is closely related to the complex Plateau problem and the problem describing the local hull of holomorphy for a real submanifold in a complex space. In [HY3], we showed that when there is at least one elliptic direction for a CR singular point in a codimension two real analytic submanifold M , it can always be holomorphically flattened near p and bound a Levi-flat hypersurface serving as the local hull of holomorphy of the manifold M near p. In this article, we will show that even if there is no elliptic direction, whenever not all Bishop invariants are parabolic, the manifold M can be formally flattened. Of course, this is all we might expect due to examples by Moser-Webster [MW] even in the C 2 setting.…”

Section: Introductionmentioning

confidence: 84%

“…The general holomorphic (or, formal) flattening problem is then to ask when M can be transformed, by a biholomorphic (formal equivalence, respectively) mapping, to an open piece of the standard Levi-flat hyperplane (C n × R 1 ) × {0} ⊂ C n+1 . While the holomorphic flattening requires certain convexity (ellipticity) as in [HY3], the formal flattening can be done in a much more general setting as we will see in this paper.…”

Section: Introductionmentioning

confidence: 99%

“…However, as demonstrated even in the two dimensional case by Moser-Webster [MW] and Gong , one could not expect the holomorphic flattening in the above theorem unless some convexity is present. (See [HY3]).…”

Section: Introductionmentioning

confidence: 99%

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

Huang

Yin

2017

Advances in Mathematics

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This is the second article of the two papers, in which we investigate the holomorphic and formal flattening problem of a non-degenerate CR singular point of a codimension two real submanifold in C n with n ≥ 3. The problem is motivated from the study of the complex Plateau problem that looks for the Levi-flat hypersurface bounded by a given real submanifold and by the classical complex analysis problem of finding the local hull of holomorphy of a real submanifold in a complex space. The present article is focused on non-degenerate flat CR singular points with at least one non-parabolic Bishop invariant. We will solve the formal flattening problem in this setting. The results in this paper and those in [HY3] are taken from our earlier arxiv post [HY4]. We split [HY4] into two independent articles to avoid it being too long.

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Section: Introductionmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 99%

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

Huang

Yin

2017

Advances in Mathematics

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“…The local structure up to quadratic part in dimension n = 2 has been studied by Coffman [5], and up to isotopy by the author in [19]. For n > 1, some normal forms have been obtained recently by Yin and Huang in [14,15] and Burcea [2]. The analogous cancellation theorem to the one for n = 1 was proved in [20].…”

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confidence: 95%

On Complex Points of Codimension 2 Submanifolds

Slapar

2015

J Geom Anal

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“…Our paper takes up a very classical problem with a new tool, and gives a formal normal form for Levi-nondegenerat real analytic manifolds which under a rather simple condition (see (85)) can be shown to be convergent. Recent advances in normal forms for real submanifolds of complex spaces with respect to holomorphic transformations have been significant: We would like to cite in this context the recent works of Huang and Yin [HY09, HY16,HY17], the second author and Gong [GS16], and Gong and Lebl [GL15].…”

Section: Introductionmentioning

confidence: 99%

Convergence of the Chern–Moser–Beloshapka normal forms

Lamel

Stolovitch

2019

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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In this article, we first describe a normal form of real-analytic, Levi-nondegenerate submanifolds of C N of codimension d ≥ 1 under the action of formal biholomorphisms, that is, of perturbations of Levi-nondegenerate hyperquadrics. We give a sufficient condition on the formal normal form that ensures that the normalizing transformation to this normal form is holomorphic. We show that our techniques can be adapted in the case d = 1 in order to obtain a new and direct proof of Chern-Moser normal form theorem.The typical model for this situation is a hyperquadric, that is, a manifold of the formwhere each J k is a Hermitian n × n matrix, and the conditions of nondegeneracy and full rank are expressed byand the homogeneous parts in z andz of a series Φ(z,z, u) = j,k Φ j,k (z,z, u), where Φ j,k (tz, sz, u) = t j s k Φ j,k (z,z, u); Φ j,k is said to be of type (j, k).We then say that Φ ∈ N CM if it satisfies the following (Chern-Moser) normal form conditions: Φ j,0 = Φ 0,j = 0, j ≥ 0;There are a number of aspects particular to the case d = 1 which allow Chern and Moser to construct, based on these conditions (which arise rather naturally from a linearization of the problem with respect to the ordering by type), a convergent choice of coordinates. In particular, Chern and Moser are able to restate much of their problem in terms of ODEs, which comes from the fact that there is only one transverse variable when d = 1; in particular, existence and regularity of solutions is guaranteed. In higher codimension, this changes dramatically, and we obtain systems of PDEs; neither do we a priori know that those are solvable nor do we know anything about the regularity of their solutions (should they exist). Our normal form has to take this into account. Another aspect of the problem, which really changes dramatically from the case d = 1 to d > 1, is the second line of the normal form conditions above: We cannot impose that

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

Cited by 22 publications

References 26 publications

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

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

On Complex Points of Codimension 2 Submanifolds

Convergence of the Chern–Moser–Beloshapka normal forms

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