Equivalence problem for Bishop surfaces

Huang, Xiaojun; Yin, Wanke

doi:10.1007/s11425-010-0047-1

Cited by 4 publications

(2 citation statements)

References 36 publications

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“…Such surfaces are called Bishop surfaces and the equivalence problem for such surfaces has been studied by a number of authors. We refer the reader to the surveys [H04,HY10] for more detailed account on this topic. We shall now focus on real submanifolds for which CR singularities do not appear.…”

Section: Formal K-equivalence and First Results Let M Mmentioning

confidence: 99%

Artin’s approximation theorems and Cauchy-Riemann geometry

Mir

2014

Methods and Applications of Analysis

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Abstract. Artin's approximation theorems are powerful tools in analytic and algebraic geometry for finding solutions of systems of analytic or algebraic equations whenever a given formal solution exists. In this survey article we describe the recent developments involving the use of Artin's approximation theorems in some problems arising from Cauchy-Riemann geometry. The solution to such problems simultaneously lead to a number of results that can be stated as PDE versions of Artin's approximation theorems. The article is intended to a non-expert audience. A number of examples and open problems are also mentioned.

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Section: Formal K-equivalence and First Results Let M Mmentioning

confidence: 99%

Artin’s approximation theorems and Cauchy-Riemann geometry

Mir

2014

Methods and Applications of Analysis

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“…For Levi-degenerate hypersurfaces in C N , N ≥ 3 satisfying certain special conditions (in addition to the Hormander-Kohn bracket-generating condition) normal form constructions were carried out by Ebenfelt [16,15] and by Kossovskiy-Zaitsev [33]. For results on normal forms for real submanifolds of higher codimension as well as CR-singular submanifolds we refer to Ezhov-Schmalz [18], Beloshapka [4], Lamel-Stolovitch [36], Moser-Webster [39], Huang-Yin [20,21], Gong [19], Coffman [13], Burcea [8]. See also Zaitsev [48] for normal forms in the non-integrable setting.…”

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The equivalence theory for infinite type hypersurfaces in $\mathbb C^2$

Ebenfelt¹,

Kossovskiy²,

Lamel³

2018

Preprint

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We develop a classification theory for real-analytic hypersurfaces in C 2 in the case when the hypersurface is of infinite type at the reference point. This is the remaining, not yet understood case in C 2 in the Problème local, formulated by H. Poincaré in 1907 and asking for a complete biholomorphic classification of real hypersurfaces in complex space. One novel aspect of our results is a notion of smooth normal forms for real-analytic hypersurfaces. We rely fundamentally on the recently developed CR -DS technique in CR-geometry.

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A normal form for a real 2-codimensional submanifold inCN+1near a CR singularity

Burcea

2013

Advances in Mathematics

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We construct a formal normal form for a class of real 2-codimensional submanifolds M ⊂ C N +1 defined near a CR singularity approximating the sphere. Our result gives a generalization of Huang-Yin's normal form in C 2 to a higher dimensional analogue case. Keywords: normal form, CR singularity, Fischer decompositionwhere ϕ ′ m,n z ′ , z ′ is a bihomogeneous polynomial of bidegree (m, n) in z ′ , z ′ satisfying the following normalization conditionsA few words about the construction of the normal form. We want to find a formal biholomorphic map sending M into a formal normal form. This leads us to study an infinite system of homogeneous equations by truncating the original equation. As in the paper [14] of Huang-Yin, this system is a semi-non linear system and is very hard to solve. We have then to use the powerful Huang-Yin's strategy and defining the weight of z k to be 1 and the weight of z k to be s − 1, for all k = 1 . . . , N . Since Aut 0 (M∞) is infinite-dimensional, it follows that the homogeneous linearized normalization equations (see sections 3 and 4) have nontrivial kernel spaces. By using the preceding system of weights and a similar argument as in the paper [14] of Huang-Yin , we are able to trace precisely how the lower order terms arise in non-linear fashion: The kernel space of degree 2t + 1 is restricted by imposing a normalization condition on ϕ ′ ts+1,0 (z) and the kernel space of degree 2t + 2 by imposing normalization conditions on ϕ ′ ts,0 (z). The non-uniqueness part of the lower degree solutions are uniquely determined in the higher order equations.

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Equivalence problem for Bishop surfaces

Cited by 4 publications

References 36 publications

Artin’s approximation theorems and Cauchy-Riemann geometry

Artin’s approximation theorems and Cauchy-Riemann geometry

The equivalence theory for infinite type hypersurfaces in $\mathbb C^2$

A normal form for a real 2-codimensional submanifold inCN+1near a CR singularity

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