On the convergence of formal mappings

Mir, Nordine

doi:10.4310/cag.2002.v10.n1.a3

Cited by 18 publications

(19 citation statements)

References 22 publications

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“…This kind of parameter dependence result can not be obtained by using the techniques of [2,7]. It is obtained as a combination of some previous results of the second author [17,18] and a key algebraic property, Proposition 3.1, proved in Section 3.…”

Section: Theorem 13 Let (M P) Be a Germ Of A Holomorphically Nondesupporting

confidence: 55%

Holomorphic versus algebraic equivalence for deformations of real-algebraic Cauchy–Riemann manifolds

Lamel

Mir

2010

Communications in Analysis and Geometry

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Section: Theorem 13 Let (M P) Be a Germ Of A Holomorphically Nondesupporting

confidence: 55%

Holomorphic versus algebraic equivalence for deformations of real-algebraic Cauchy–Riemann manifolds

Lamel

Mir

2010

Communications in Analysis and Geometry

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“…We mention here work of Zaitsev [Z97]- [Z99], joint work of the first and third authors with Zaitsev [BRZ00], Mir [Mi00a]- [Mi00b], as well as work of the authors [BER96], [BER98], [BER99b], [BER99c]. We should also mention the work of Christ-NagelStein-Wainger [CNSW99] in which the ranks of iterated real mappings are related to curvature and finite type conditions in a very different context.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of the Segre varieties of a real submanifold in complex space

2002

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For a smooth (or formal) generic submanifold M M of real codimension d d in complex space C N \mathbb {C}^N with 0 ∈ M 0\in M , we introduce the notion of a formal Segre variety mapping γ : ( C N × C N − d , 0 ) → ( C N , 0 ) \gamma : (\mathbb {C}^N\times \mathbb {C} ^{N-d},0)\to (\mathbb {C}^N,0) and its iterated Segre mappings at 0 0 , v j : ( C ( N − d ) j , 0 ) → ( C N , 0 ) v^j:(\mathbb {C}^{(N-d)j},0) \to (\mathbb {C}^N,0) , j ≥ 1 j\ge 1 . The Segre variety mapping γ \gamma extends the notion of Segre varieties of a real-analytic generic submanifold to the setting of smooth (or formal) submanifolds. One of the main results in this paper is that M M is of finite type (in the sense of Kohn and Bloom–Graham) at 0 0 if and only if there exists k 0 ≤ d + 1 k_0\le d+1 such that the (generic) rank of v k 0 v^{k_0} is N N . More generally, we prove that v k 0 v^{k_0} parameterizes the local CR orbit of M M at 0 0 .

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“…As matters stand, such a kind of characterization for the regularity of CR maps happens to be known already in case where at least one of the two hypersurfaces is algebraic (cf. [5,6,12]). In fact, in the algebraic case, one can apply the classical "polynomial identities" in the spirit of Baouendi-Jacobowitz-Treves.…”

Section: Identity ρ (H(t)h(t)) ≡ B(tt)ρ(tt)mentioning

confidence: 99%

Convergence of formal invertible CR mappings between minimal holomorphically nondegenerate real analytic hypersurfaces

Merker

2001

International Journal of Mathematics and Mathematical Sciences

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Abstract. Recent advances in CR (Cauchy-Riemann) geometry have raised interesting fine questions about the regularity of CR mappings between real analytic hypersurfaces. In analogy with the known optimal results about the algebraicity of holomorphic mappings between real algebraic sets, some statements about the optimal regularity of formal CR mappings between real analytic CR manifolds can be naturally conjectured. Concentrating on the hypersurface case, we show in this paper that a formal invertible CR mapping between two minimal holomorphically nondegenerate real analytic hypersurfaces in C n is convergent. The necessity of holomorphic nondegeneracy was known previously. Our technique is an adaptation of the inductional study of the jets of formal CR maps which was discovered by Baouendi-Ebenfelt-Rothschild. However, as the manifolds we consider are far from being finitely nondegenerate, we must consider some new conjugate reflection identities which appear to be crucial in the proof. The higher codimensional case will be studied in a forthcoming paper.2000 Mathematics Subject Classification. 32V25, 32V35, 32V40.

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On the convergence of formal mappings

Cited by 18 publications

References 22 publications

Holomorphic versus algebraic equivalence for deformations of real-algebraic Cauchy–Riemann manifolds

Holomorphic versus algebraic equivalence for deformations of real-algebraic Cauchy–Riemann manifolds

Dynamics of the Segre varieties of a real submanifold in complex space

Convergence of formal invertible CR mappings between minimal holomorphically nondegenerate real analytic hypersurfaces

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