On wedge extendability of CR-meromorphic functions

Abstract

In this article, we consider metrically thin singularities E of the solutions of the tangential Cauchy-Riemann operators on a C 2,α -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 \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. Subject Classification (1991): 32D20, 32A20, 32D10, 32V10, 32V25, 32V35 Mathematics

Analytic continuation of holomorphic mappings

“…Supposons que l'ensemble des points limites cl f (M) est contenu dans une hypersurface, fermée, lisse, algébrique réelle M ⊂ U de type fini, où U est un ouvert de C n . Si f se prolonge continûment sur une partie ouverte de M, alors elle se prolonge holomorphiquement au voisinage de chaque point de M.La preuve est basée sur la propagation de l'analycité des applications holomorphes à travers les variétés de Segre et sur la construction d'une famille d'ellipsoïdes utilisée dans [6]. D'abord, nous montrons la proposition suivante :…”

“…La preuve est basée sur la propagation de l'analycité des applications holomorphes à travers les variétés de Segre et sur la construction d'une famille d'ellipsoïdes utilisée dans [6]. D'abord, nous montrons la proposition suivante :…”

On Boundary Regularity of Holomorphic Correspondences

“…Note that we do not require pseudoconvexity of M or M ′ . The proof of Theorem 1 is based on the idea of analytic continuation of holomorphic correspondences along Segre varieties and the construction of a family of ellipsoids used in [14]. Theorem 1 is already known if M and M ′ are closed smooth real-analytic hypersurfaces of finite type and in additional f is continuous on M ( [23]).…”

“…Since M is globally minimal, there exists a CR-curve γ (i.e., the tangent vector to γ at any point is contained in the complex tangent to M ) passing through q and entering M h . We will use the construction of a family of ellipsoids used by Merker and Porten [14]. Figure 1.…”

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

“…This family Aql will be our starting point to study the envelope of holomorphy of (a certain subdomain of) the union of D together with a neighborhood Q of M1 and with an arbitrarily thin neighborhood of (see Figure 3 To deform these discs by applying the classical works on analytic discs and because Banach spaces are necessary, we shall work in the regularity class ck,a, where k > 1 is arbitrary and where 0 a 1, which is sufficient for our purposes. Let T1 denote the Hilbert transform vanishing at 1 (see [Tul], [Tu2], [Tu3], [MP1], [MP2] depending only on the Jacobian matrix of the mapping in 3) at 0 x 1 such that c-1 b ~ c6. Let 0 ( 1, b) denote the disc of radius 6 centered at 1 E C. Furthermore, since the boundary of the disc ApI,o is transversal to then after shrinking a bit "7 if necessary, we can assume that the set contains and foliates by half analytic discs the whole lower side An (0, 2TI) n E" (see Figure 6).…”