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53
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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327
Sep 18, 2013
09/13
by
Joël Merker; Egmont Porten
texts
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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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79
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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109
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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72
Sep 18, 2013
09/13
by
Joel Merker; Egmont Porten
texts
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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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54
Sep 18, 2013
09/13
by
Joel Merker; Egmont Porten
texts
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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
