scholarly journals On holomorphic extension of functions on singular real hypersurfaces inℂn

2001 ◽  
Vol 26 (3) ◽  
pp. 173-178
Author(s):  
Tejinder S. Neelon
Keyword(s):  
Limit Point ◽  
Infinite Order ◽  
Holomorphic Extension ◽  
Real Hypersurfaces ◽  
Zero Set ◽  
Riemann Equation ◽  
Real Analytic ◽  
Cauchy Riemann ◽  
Extension Of Functions ◽  
Tangential Cauchy Riemann Equation

The holomorphic extension of functions defined on a class of real hypersurfaces inℂnwith singularities is investigated. Whenn=2, we prove the following: everyC1function onΣthat satisfies the tangential Cauchy-Riemann equation on boundary of{(z,w)∈ℂ2:|z|k<P(w)},P∈C1,P≥0andP≢0, extends holomorphically inside provided the zero setP(w)=0has a limit point orP(w)vanishes to infinite order. Furthermore, ifPis real analytic then the condition is also necessary.

scholarly journals ON THE CLASSIFICATION BY MORIMOTO AND NAGANO

2019 ◽  
pp. 1-13
Author(s):  
ALEXANDER ISAEV
Keyword(s):  
Three Dimensional ◽  
Parameter Range ◽  
Real Hypersurfaces ◽  
Polynomial Map ◽  
Real Analytic ◽  
Cauchy Riemann ◽  
Real Analytic Hypersurfaces ◽  
Simply Connected

We consider a family $M_{t}^{3}$ , with $t>1$ , of real hypersurfaces in a complex affine three-dimensional quadric arising in connection with the classification of homogeneous compact simply connected real-analytic hypersurfaces in $\mathbb{C}^{n}$ due to Morimoto and Nagano. To finalize their classification, one needs to resolve the problem of the Cauchy–Riemann (CR)-embeddability of $M_{t}^{3}$ in $\mathbb{C}^{3}$ . In our earlier article, we showed that $M_{t}^{3}$ is CR-embeddable in $\mathbb{C}^{3}$ for all $1<t<\sqrt{(2+\sqrt{2})/3}$ . In the present paper, we prove that $M_{t}^{3}$ can be immersed in $\mathbb{C}^{3}$ for every $t>1$ by means of a polynomial map. In addition, one of the immersions that we construct helps simplify the proof of the above CR-embeddability theorem and extend it to the larger parameter range $1<t<\sqrt{5}/2$ .

scholarly journals Global Regularity for the∂-b-Equation onCRManifolds of Arbitrary Codimension

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Shaban Khidr ◽  
Osama Abdelkader
Keyword(s):  
Existence Theorem ◽  
Complex Manifold ◽  
Global Regularity ◽  
Riemann Equation ◽  
Complex Dimension ◽  
Arbitrary Codimension ◽  
Cauchy Riemann ◽  
Tangential Cauchy Riemann Equation

LetMbe aC∞compactCRmanifold ofCR-codimensionl≥1andCR-dimensionn-lin a complex manifoldXof complex dimensionn≥3. In this paper, assuming thatMsatisfies conditionY(s)for someswith1≤s≤n-l-1, we prove anL2-existence theorem and global regularity for the solutions of the tangential Cauchy-Riemann equation for(0,s)-forms onM.

Holomorphic Immersions of Bi-Disks into 9D Real Hypersurfaces with Levi Signature (2,2)

2020 ◽  
Author(s):  
Wei Guo Foo ◽  
Joël Merker
Keyword(s):  
Fixed Point ◽  
Real Hypersurface ◽  
Real Hypersurfaces ◽  
Zero Set ◽  
Cr Manifolds ◽  
Real Analytic ◽  
Unit Disks ◽  
Complex Valued

Abstract Inspired by an article of R. Bryant on holomorphic immersions of unit disks into Lorentzian CR manifolds, we discuss the application of Cartan’s method to the question of the existence of bi-disk $\mathbb{D}^{2}$ in a smooth $9$D real-analytic real hypersurface $M^{9}\subset \mathbb{C}^{5}$ with Levi signature $(2,2)$ passing through a fixed point. The result is that the lift to $M^{9}\times U(2)$ of the image of the bi-disk in $M^{9}$ must lie in the zero set of two complex-valued functions in $M^{9}\times U(2)$. We then provide an example where one of the functions does not identically vanish, thus obstructing holomorphic immersions.

SUR L'ÉQUATION DE CAUCHY–RIEMANN TANGENTIELLE DANS UNE CALOTTE STRICTEMENT PSEUDOCONVEXE

2005 ◽  
Vol 16 (09) ◽  
pp. 1063-1079 ◽  
Author(s):  
CHRISTINE LAURENT-THIÉBAUT
Keyword(s):  
Stein Manifold ◽  
Geometrical Characterization ◽  
Riemann Equation ◽  
Pseudoconvex Boundary ◽  
Cauchy Riemann ◽  
Tangential Cauchy Riemann Equation

We search a cohomological and a geometrical characterization of the open subsets of a strictly pseudoconvex boundary in a Stein manifold on which one can solve the tangential Cauchy–Riemann equation in all bidegrees. On cherche une caractérisation cohomologique et géométrique des ouverts du bord d'un domaine strictement pseudoconvexe relativement compact d'une variété de Stein sur lesquels on peut résoudre l'équation de Cauchy–Riemann tangentielle en tout bidegré.

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