27 research outputs found
On removable singularities for integrable CR functions
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
Let be a generic CR submanifold in \C^{m+n}, ,, . A CR meromorphic mapping (in the sense
of Harvey-Lawson) is a triple , where: 1. is a -smooth mapping defined over a dense open subset
of with values in a projective manifold ; 2. The closure
of its graph in \C^{m+n} \times Y defines a oriented scarred
-smooth CR manifold of CR dimension (i.e. CR outside a closed
thin set) and 3. Such that in the sense of currents. We prove
in this paper that extends meromorphically to a
wedge attached to if is everywhere minimal and
(real analytic) or if is a globally minimal
hypersurface.Comment: 25 pages, LaTeX. To appear in Ann. Pol. Math. 199
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
Characteristic foliations on maximally real submanifolds of C^n and envelopes of holomorphy
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
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 ,
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
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
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