On envelopes of holomorphy of domains covered by Levi-flat hats and the reflection principle

Merker, Joël

doi:10.5802/aif.1922

Cited by 8 publications

(6 citation statements)

References 55 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Thanks to this proposition, in order to characterize the sphericality of a local real analytic hypersurface M ⊂ C 2 explicitly in terms of its complex defining function Θ, our strategy 15 will be to: characterize the local equivalence to w ′ z ′ z ′ (z ′ ) = 0 of the associated differential equation: (6) w zz (z) = Θ zz z, ζ z, w(z), w z (z) , ξ z, w(z), w z (z) , explicitly in terms of the three functions Θ zz , ζ and ξ; eliminate any occurence of the two auxiliary functions ζ and ξ so as to re-express the obtained result only in terms of the sixth-order jet J 6 z,z,w Θ. §3.…”

Section: )mentioning

confidence: 99%

“…On the other hand, in the studies [13,14,15,16,17] devoted to the CR reflection principle, it was emphasized that all the adequate invariants of CR mappings between CR manifolds: Pair of Segre foliations, Segre chains, Complexified CR orbits, Jets of complexified Segre varietes, Rigidity of formal CR mappings, Nondegeneracy conditions, CR-reflection function 1 , can be viewed correctly only when M is represented by a so-called complex defining equation of the form:…”

Section: §1 Introductionmentioning

confidence: 99%

“…For a presentation of these concepts, the reader is referred to the extensive introductions of[15,17] and also to[20] for more about why dealing only with complex defining equations is natural and unavoidable when one wants to insert CR geometry in the wider universe of completely integrable systems of real or complex analytic partial differential equations 2. More will be said shortly in Section 2 below.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Nonrigid spherical real analytic hypersurfaces in ℂ²

Merker¹

2010

Complex Variables and Elliptic Equations

Self Cite

View full text Add to dashboard Cite

A Levi nondegenerate real analytic hypersurface M of C 2 represented in local coordinates (z, w) ∈ C 2 by a complex defining equation of the form w = Θ(z, z, w) which satisfies an appropriate reality condition, is spherical if and only if its complex graphing function Θ satisfies an explicitly written sixth-order polynomial complex partial differential equation. In the rigid case (known before), this system simplifies considerably, but in the general nonrigid case, its combinatorial complexity shows well why the two fundamental curvature tensors constructed by Élie Cartan in 1932 in his classification of hypersurfaces have, since then, never been reached in parametric representation.

show abstract

Section: )mentioning

confidence: 99%

Section: §1 Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

Nonrigid spherical real analytic hypersurfaces in ℂ²

Merker¹

2010

Complex Variables and Elliptic Equations

Self Cite

View full text Add to dashboard Cite

show abstract

“…Assume a para-CR equivalence (1.2) is given. In the less general CR context, the following theorem already appeared in [30,38,37], and in Section 12, Theorem 12.3 will provide more details.…”

Section: Introductionmentioning

confidence: 99%

“…In a CR context, partial aspects of the topics presented in this Section 12 and in the next Sections 13, 14, 15, 16, 17, have already been studied in[30,38,37].…”

mentioning

confidence: 99%

Equivalences of PDE systems associated to degenerate para-CR Structures: foundational aspects

Merker¹

2021

Preprint

View full text Add to dashboard Cite

We study basic invariants of submanifolds of solutions) . Two Levi forms exist, and have the same rank r min(n, m). If M is k-nondegenerate with respect to parameters and l-nondegenerate with respect to variables, Aut(M ) is a local Lie group of dimension: dim Aut(M ) n+1+2k+2l 2k+2lmin (n + 1), (m + 1) .Mainly, our goal is to set up foundational material addressed to CR geometers. We focus on n = m = 2, assuming r = 1. In coordinates (x, y, z, a, b, c), a local equation is:with β and β representing the two 2-nondegeneracy invariants at 0. The associated para-CR PDE system:

show abstract

Some regularity theorems for CR mappings

Balakumar

Verma

2012

Math. Z.

View full text Add to dashboard Cite

The purpose of this article is to study Lipschitz CR mappings from an h-extendible (or semi-regular) hypersurface in C n . Under various assumptions on the target hypersurface, it is shown that such mappings must be smooth. A rigidity result for proper holomorphic mappings from strongly pseudoconvex domains is also proved.Abstract. The purpose of this article is to study Lipschitz CR mappings from an h-extendible (or semi-regular) hypersurface in C n . Under various assumptions on the target hypersurface, it is shown that such mappings must be smooth. A rigidity result for proper holomorphic mappings from strongly pseudoconvex domains is also proved.

show abstract

On envelopes of holomorphy of domains covered by Levi-flat hats and the reflection principle

Cited by 8 publications

References 55 publications

Nonrigid spherical real analytic hypersurfaces in ℂ²

Nonrigid spherical real analytic hypersurfaces in ℂ²

Equivalences of PDE systems associated to degenerate para-CR Structures: foundational aspects

Some regularity theorems for CR mappings

Contact Info

Product

Resources

About

On envelopes of holomorphy of domains covered by Levi-flat hats and the reflection principle

Cited by 8 publications

References 55 publications

Nonrigid spherical real analytic hypersurfaces in ℂ2

Nonrigid spherical real analytic hypersurfaces in ℂ2

Equivalences of PDE systems associated to degenerate para-CR Structures: foundational aspects

Some regularity theorems for CR mappings

Contact Info

Product

Resources

About

Nonrigid spherical real analytic hypersurfaces in ℂ²

Nonrigid spherical real analytic hypersurfaces in ℂ²