27 research outputs found

    On removable singularities for integrable CR functions

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    We endeavour a systematic approach for the removal of singularities for CR functions on an arbitrary embeddable CR manifold.Comment: 38 pages, LaTeX. To appear in Indiana Univ. Math. J. 199

    On the local meromorphic extension of CR meromorphic mappings

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    Let MM be a generic CR submanifold in \C^{m+n}, m=CRdimM1m= CRdim M \geq 1,n=codimM1n=codim M \geq 1, d=dimM=2m+nd=dim M = 2m+n. A CR meromorphic mapping (in the sense of Harvey-Lawson) is a triple (f,Df,[Γf])(f,{\cal D}_f, [\Gamma_f]), where: 1. f:DfYf: {\cal D}_f \to Y is a C1{\cal C}^1-smooth mapping defined over a dense open subset Df{\cal D}_f of MM with values in a projective manifold YY; 2. The closure Γf\Gamma_f of its graph in \C^{m+n} \times Y defines a oriented scarred C1{\cal C}^1-smooth CR manifold of CR dimension mm (i.e. CR outside a closed thin set) and 3. Such that d[Γf]=0d[\Gamma_f]=0 in the sense of currents. We prove in this paper that (f,Df,[Γf])(f,{\cal D}_f, [\Gamma_f]) extends meromorphically to a wedge attached to MM if MM is everywhere minimal and Cω{\cal C}^{\omega} (real analytic) or if MM is a C2,α{\cal C}^{2,\alpha} globally minimal hypersurface.Comment: 25 pages, LaTeX. To appear in Ann. Pol. Math. 199

    Metrically thin singularities of integrable CR functions

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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

    Characteristic foliations on maximally real submanifolds of C^n and envelopes of holomorphy

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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 problem by means of purely local techniques, namely by means of families of small analytic discs partially attached to maximally real submanifolds of C^n and by means of a thorough study of the relative disposition of the characteristic foliation with respect to the track on M of a certain half-wedge attached to M. This localization procedure enables us to answer an open problem raised by B. J\"oricke: under a certain nontransversality condition with respect to the characteristic foliation, we show that every closed subset C of a C^{2,a}-smooth maximally real submanifold M^1 of a (n-1)-codimensional generic C^{2,a}-smooth submanifold of \C^n is CR-, W- and L^p-removable (Theorem~1.2'). The known removability results in CR dimension at least two appear to be logical consequences of Theorem~1.2'. The main proof (65p.) is written directly in arbitrary codimension. Finally, we produce an example of a nonremovable 2-torus contained in a maximally real 3-dimensional maximally real submanifold, showing that the nontransversality condition is optimal for universal removability. Numerous figures are included to help readers who are not insiders of higher codimensional geometry.Comment: 113 pages, 24 figures, LaTe

    Truncated tube domains with multi-sheeted envelope

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    The present article is concerned with a group of problems raised by J. Noguchi and M. Jarnicki/P. Plug, namely whether the envelopes of holomorphy of truncated tube domains are always schlicht, i.e. subdomains of Cn\mathbb{C^n}, and how to characterize schlichtness if this is not the case. By way of a counter-example homeomorphic to the 4-ball, we answer the first question in the negative. Moreover, it is possible that the envelopes have arbitrarily many sheets. The article is concluded by sufficient conditions for schlichtness in complex dimension two.Comment: 11 pages, 3 figure

    Holomorphic extension of CR functions, envelopes of holomorphy and removable singularities

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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 orbit; propagation of CR extendability and edge-of-the-wedge theorem; Painlev\'{e} problem; metrically thin singularities of CR functions; geometrically removable singularities for solutions of the induced d-barre. Selected theorems are fully proved, while surveyed results are put in the right place in the architecture.Comment: 283 pages ; 33 illustrations ; 16 open problems http://www.hindawi.com/journals/imrs

    Wedge extendability of CR-meromorphic functions: the minimal case

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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 globally minimal. As an application, we deduce that there exists a wedgelike domain attached to an everywhere locally minimal M to which every CR-meromorphic function on M extends meromorphically
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