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Joel Merker

Publications and source records attributed to Joel Merker.

66 records · Page 4Linked to original sources
arXiv2004

Wedge extendability of CR-meromorphic functions: the minimal case

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.

math.CV↗
arXiv2004

On the partial algebraicity of holomorphic mappings between real algebraic sets

In this paper, we consider local holomorphic mappings f: M\to M' between real algebraic CR generic manifolds (or more generally, real algebraic sets with singularities) in the complex euclidean spaces of different dimensions and we search necessary and sufficient conditions for f to be algebraic. These conditions appear to exclude two certain flatness of M and of M'. From the point of view of CR geometry, a real analytic CR manifold M can be flat in essentially to ways, being biholomorphic to a product M_1\times Δ^k, k\geq 1, by a polydisc (algebraic degeneracy), or to a product M_1\times I^l,l\geq 1 by a real cube (transversal degeneracy), in a neighborhod of a Zariski generic point. We also require that the CR manifold M is minimal in the sense of Tumanov at a generic point. Our first result provides a characterization of mappings with positive transcendence degree k. Such maps have the property that near a Zariski generic point f(p) in M', there exists a k-algebraically degenerate real algebraic set X'' which contains f(M) and which is contained in M'. This solves the algebraic mapping problem for a minimal source M completely. Our second main result is the construction of canonical foliations by Segre surfaces of the extrinsic complexification of M. We prove in particular that a holomorphic function defined in a neighborhood of a minimal M is algebraic if and only if its restriction to each Segre surface of M is algebraic. We also show by an example that the double reflection foliation in the spirit of tangential CR derivations does not yield a characterization of positivity of transcendence degree of holomorphic mappings.

math.CV↗
arXiv2004

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

In the present paper, we associate the techniques of the Lewy-Pinchuk reflection principle with the Behnke-Sommer continuity principle. Extending a so-called reflection function to a parameterized congruence of Segre varieties, we are led to studying the envelope of holomorphy of a certain domain covered by a smooth Levi-flat ``hat''. In our main theorem, we show that every C^infty-smooth CR diffeomorphism h: M to M' between two globally minimal real analytic hypersurfaces in C^n (n >1) is real analytic at every point of M if M' is holomorphically nondegenerate. More generally, we establish that the reflection function R_h' associated to such a C^infty-smooth CR diffeomorphism between two globally minimal hypersurfaces in C^n (n > 1) always extends holomorphically to a neighborhood of the graph of \bar{h} in M\times \bar{M}', without any nondegeneracy condition on M'. This gives a new version of the Schwarz symmetry principle in several complex variables. Finally, as an appendix, we show that every C^infty-smooth CR mapping h: M to M' between two real analytic hypersurfaces containing no complex curves is real analytic at every point of M, without any rank condition on h. This answers a conjecture, explicitely stated for C^infty-smooth maps at p.~328 of the survey article: S.M. Baouendi, P. Ebenfelt and L.-P. Rothschild, Bull. Amer. Math. Soc. (N.S.) 37 (2000), no. 3, 309--336. Importantly, notice that a much stronger result, valid for continuous maps, has been obtained by K. Diederich and S. Pinchuk in: Michigan Math. J. {\bf 51} (2003), no. 1, 111--140; no.~3, 667--668.

math.CV↗
arXiv2004

On the Hartogs-Bochner phenomenon for CR functions in P_2(C)

Let M be a compact, connected, C^2-smooth and globally minimal hypersurface M in P_2(C) which divides the projective space into two connected parts U^{+} and U^{-}. We prove that there exists a side, U^- or U^+, such that every continuous CR function on M extends holomorphically to this side. Our proof of this theorem is a simplification of a result originally due to F. Sarkis.

math.CV↗
arXiv2004

On the local algebraizability of real analytic generic submanifolds of C^n

We prove that the local (pseudo)group of biholomorphisms stabilizing a minimal, finitely nondegenerate real algebraic submanifold in C^n is a real algebraic local Lie group (the works of S.M. Baouendi, P. Ebenfelt, L.-P. Rothschild and D. Zaitsev published in this direction restrict, without understandable reason, to the isotropy group of a fixed central point). We deduce necessary conditions for the local algebraizability of real analytic rigid tubes of arbitrary codimension in C^n. Without using the Elie Cartan equivalence algorithm, we explain the up to now only known example Im w = e^(|z^2|), due to X. Huang, S. Ji and S.S. Yau in 2001, of a nonalgebraizable Levi nondegenerate real analytic hypersurface of C^2. These elementary criteria provide a first answer to an open problem raised by S.M. Baouendi, P. Ebenfelt and L.-P. Rothschild, in the survey article : ``Local geometric properties of real submanifolds in complex space'', Bull. Amer. Math. Soc. (N.S.) 37 (2000), no. 3, 309--336. A second answer (necessary and sufficient condition) for local algebraizability in the homogeneous case will appear subsequently.

math.CV↗
arXiv2000

Vector field construction of Segre sets

A CR generic real analytic CR manifold M carries two families of Segre varieties and conjugate Segre varieties. We observe in this article that their complexifications give rise to two families of foliations of the complexification of M which coincide with the flow foliations induced by the complexified CR (1,0) and (0,1) vector fields tangent to M. As an application, we derive a new proof of the characterization of finite type in terms of Segre sets.

math.CV↗
arXiv1999

Metrically thin singularities of integrable CR functions

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.

math.CV↗