Normal forms for CR singular codimension-two Levi-flat submanifolds

Gong, Xianghong; Lebl, Jiri

doi:10.2140/pjm.2015.275.115

Cited by 15 publications

(12 citation statements)

References 30 publications

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“…The author proved [8] a similar result when the model (1.3) is perturbated by terms of degree 3. Motivated by Gong-Lebl [20], his approach [8] was based on trying to understand the C.-R. structure existent near a C.-R. Singularity. That approach [8] does not apply in this situation (1.1).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

CR Singularities and Generalizations of Moser's Theorem II

Burcea¹

2019

Preprint

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We study the local equivalence problems for real-analytic manifold M which is formally (biholomorphically) equivalent to the model, where < 2λ 1 , . . . , 2λ N < 1. It is proven that M is biholomorphically equivalent to this model by developing a partial normal form for such real-formal submanifold using generalized Fischer Decompositions. These methods allow also to develop also a (formal) normal form for a class of C.-R. Singular 2-Codimensional Real Submanifolds in Complex Spaces.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

CR Singularities and Generalizations of Moser's Theorem II

Burcea¹

2019

Preprint

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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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“…The work of Bishop in C 2 , especially in the elliptic case (λ < 1 2 ), has been refined by Kenig-Webster [18], Moser-Webster [22], Moser [21], Huang-Krantz [13], and many others, see for example Huang-Yin [14] and the references therein for recent work. For work in higher dimensions, especially in codimension two, see Huang-Yin [15][16][17], Gong-Lebl [10], Burcea [5,6], Dolbeault-Tomassini-Zaitsev [8,9], Coffman [7], Slapar [25], and the authors themselves [20].…”

Section: Introductionmentioning

confidence: 99%

Codimension Two CR Singular Submanifolds and Extensions of CR Functions

2017

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Abstract. Let M ⊂ C n+1 , n ≥ 2, be a real codimension two CR singular real-analytic submanifold that is nondegenerate and holomorphically flat. We prove that every realanalytic function on M that is CR outside the CR singularities extends to a holomorphic function in a neighborhood of M . Our motivation is to prove the following analogue of the Hartogs-Bochner theorem. Let Ω ⊂ C n × R, n ≥ 2, be a bounded domain with a connected real-analytic boundary such that ∂Ω has only nondegenerate CR singularities. We prove that if f : ∂Ω → C is a real-analytic function that is CR at CR points of ∂Ω, then f extends to a holomorphic function on a neighborhood of Ω in C n × C.

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Normal forms for CR singular codimension-two Levi-flat submanifolds

Cited by 15 publications

References 30 publications

CR Singularities and Generalizations of Moser's Theorem II

CR Singularities and Generalizations of Moser's Theorem II

Convergence of the Chern–Moser–Beloshapka normal forms

Codimension Two CR Singular Submanifolds and Extensions of CR Functions

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