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Jul 20, 2013
07/13

by
Mauro Nacinovich; Egmont Porten

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Let $M$ be a $CR$ submanifold of a complex manifold $X$. The main result of this article is to show that $CR$-hypoellipticity at $p_0\in{M}$ is necessary and sufficient for holomorphic extension of all germs of $CR$ functions to an ambient neighborhood in $X$. As an application, we obtain that $CR$-hypoellipticity implies the existence of generic embeddings and prove holomorphic extension for a large class of $CR$ manifolds satisfying a higher order Levi pseudoconcavity condition.

Source: http://arxiv.org/abs/1107.3374v2

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Sep 19, 2013
09/13

by
Joël Merker; Egmont Porten

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100 years ago exactly, in 1906, Hartogs published a celebrated extension phenomenon (birth of Several Complex Variables), whose global counterpart was stated in full generality later by Osgood (1929): holomorphic functions in a connected neighborhood V(bD) of a connected boundary bD contained in C^n (n >= 2) do extend holomorphically and uniquely to the domain D. It was a long-standing open problem to derive a proof using only analytic discs, as did Hurwitz (1897), Hartogs (1906) and E.E....

Source: http://arxiv.org/abs/math/0610985v2

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Jun 30, 2018
06/18

by
Sandra Carillo; Mauro Lo Schiavo; Egmont Porten; Cornelia Schiebold

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A noncommutative KdV-type equation is introduced extending the Baecklund chart in [S. Carillo, M. Lo Schiavo, and C. Schiebold, SIGMA 12 (2016)]. This equation, called meta-mKdV here, is linked by Cole-Hopf transformations to the two noncommutative versions of the mKdV equations listed in [P.J. Olver and V.V. Sokolov Commun. Math. Phys. 193 (1998), Theorem 3.6]. For this meta-mKdV, and its mirror counterpart, recursion operators, hierarchies and an explicit solution class are derived.

Topics: Mathematical Physics, Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1704.03208

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Sep 18, 2013
09/13

by
Joël Merker; Egmont Porten

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Let S be an arbitrary real surface, with or without boundary, contained in a hypersurface M of the complex euclidean space \C^2, with S and M of class C^{2, a}, where 0 < a < 1. If M is globally minimal, if S is totally real except at finitely many complex tangencies which are hyperbolic in the sense of E. Bishop and if the union of separatrices is a tree of curves without cycles, we show that every compact K of S is CR-, W- and L^p-removable (Theorem~1.3). We treat this seemingly global...

Source: http://arxiv.org/abs/math/0401142v1

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Jul 22, 2013
07/13

by
Andrea Altomani; C. Denson Hill; Mauro Nacinovich; Egmont Porten

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We prove a subelliptic estimate for systems of complex vector fields under some assumptions that generalize the essential pseudoconcavity for $CR$ manifolds and H\"ormander's bracket condition for real vector fields. Applications are given to prove the hypoellipticity of first order systems and second order partial differential operators. Finally we describe a class of compact homogeneous CR manifolds for which the distribution of $(0,1)$ vector fields satisfies a subelliptic estimate. v2:...

Source: http://arxiv.org/abs/0807.4857v2

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64

Jul 20, 2013
07/13

by
J. Merker; Egmont Porten

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Soient $M$ une vari\'et\'e CR localement plongeable et $\Phi\subset M$ un ferm\'e. On donne des conditions suffisantes pour que les fonctions $L_{loc}^1$ qui sont CR sur $M\backslash \Phi$ le soient aussi sur $M$ tout entier.

Source: http://arxiv.org/abs/math/9902039v1

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Sep 19, 2013
09/13

by
Joël Merker; Egmont Porten

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This is an extensive (published) survey on CR geometry, whose major themes are: formal analytic reflection principle; generic properties of Systems of (CR) vector fields; pairs of foliations and conjugate reflection identities; Sussmann's orbit theorem; local and global aspects of holomorphic extension of CR functions; Tumanov's solution of Bishop's equation in Hoelder classes with optimal loss of smoothness; wedge-extendability on C^2,a generic submanifolds of C^n consisting of a single CR...

Source: http://arxiv.org/abs/math/0701531v1

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108

Jul 20, 2013
07/13

by
Joel Merker; Egmont Porten

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In this article, we consider metrically thin singularities A of the tangential Cauchy-Riemann operator on smoothly embedded Cauchy-Riemann manifolds M. The main result states removability within the space of locally integrable functions on M under the hypothesis that the (dim M-2)-dimensional Hausdorff volume of A is zero and that the CR-orbits of M and M-A are comparable.

Source: http://arxiv.org/abs/math/9906056v1

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Sep 18, 2013
09/13

by
Mauro Nacinovich; Egmont Porten

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Let $M$ be a $CR$ submanifold of a complex manifold $X$. The main result of this article is to show that $CR$-hypoellipticity at $p_0\in{M}$ is necessary and sufficient for holomorphic extension of all germs of $CR$ functions to an ambient neighborhood in $X$. As an application, we obtain that $CR$-hypoellipticity implies the existence of generic embeddings and prove holomorphic extension for a large class of $CR$ manifolds satisfying a higher order Levi pseudoconcavity condition.

Source: http://arxiv.org/abs/1201.1704v1

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Jul 20, 2013
07/13

by
J. Merker; Egmont Porten

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We endeavour a systematic approach for the removal of singularities for CR functions on an arbitrary embeddable CR manifold.

Source: http://arxiv.org/abs/math/9902040v1

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39

Jul 20, 2013
07/13

by
J. Merker; Egmont Porten

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Let $M$ be a generic CR submanifold in $\C^{m+n}$, $m= CRdim M \geq 1$,$n=codim M \geq 1$, $d=dim M = 2m+n$. A CR meromorphic mapping (in the sense of Harvey-Lawson) is a triple $(f,{\cal D}_f, [\Gamma_f])$, where: 1. $f: {\cal D}_f \to Y$ is a ${\cal C}^1$-smooth mapping defined over a dense open subset ${\cal D}_f$ of $M$ with values in a projective manifold $Y$; 2. The closure $\Gamma_f$ of its graph in $\C^{m+n} \times Y$ defines a oriented scarred ${\cal C}^1$-smooth CR manifold of CR...

Source: http://arxiv.org/abs/math/9902038v1

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Sep 23, 2013
09/13

by
C. Denson Hill; Egmont Porten

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The H-principle, which is the analogue, for CR manifolds, of the classical Hartogs principle in several complex variables, is known to be valid in the small on a pseudoconcave CR manifold of any codimension. However it fails in the large, as has been shown by the counterexample found in [HN1]. Hence there is an underlying obstruction to the global H-principle on a pseudoconcave CR manifold. The purpose of this note is to take the first steps toward a deeper understanding of this obstruction.

Source: http://arxiv.org/abs/0710.5728v1

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Sep 18, 2013
09/13

by
Joel Merker; Egmont Porten

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Employing Morse theory for the global control of monodromy and the method of analytic discs for local extension, we establish a version of the global Hartogs extension theorem in a singular setting: for every domain D of an (n-1)-complete normal complex space X of pure dimension n >= 2 and for every compact set K in D such that D - K is connected, holomorphic or meromorphic functions in D - K extend holomorphically or meromorphically to D. Normality is an unvavoidable assumption for...

Source: http://arxiv.org/abs/0704.3216v1

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Jun 29, 2018
06/18

by
Mauro Nacinovich; Egmont Porten

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We introduce various notions of q-pseudo-concavity for abstract CR manifolds and we apply these notions to the study of hyoo-ellipticity, maximum modulus principle and Cauchy problems for CR functions.

Topics: Complex Variables, Mathematics

Source: http://arxiv.org/abs/1611.02553

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Sep 18, 2013
09/13

by
Joel Merker; Egmont Porten

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In this article, we consider metrically thin singularities E of the solutions of the tangential Cauchy-Riemann operators on a C^{2,a}-smooth embedded Cauchy-Riemann generic manifold M (CR functions on M - E) and more generally, we consider holomorphic functions defined in wedgelike domains attached to M - E. Our main result establishes the wedge- and the L^1-removability of E under the hypothesis that the (\dim M-2)-dimensional Hausdorff volume of E is zero and that M and M\backslash E are...

Source: http://arxiv.org/abs/math/0006178v2