### A remark on a theorem by Henkin and Tumanov on separately CR functions

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

We define the notion of CR equivalence for Levi-flat foliations and compare in a local setting these foliations to their linear parts. We study also the situation where the foliation has a first integral ; a condition is given so that this integral is the real part of a holomorphic function.

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 ${\mathcal{C}}^{\infty}$-smooth CR diffeomorphism $h:M\to {M}^{\text{'}}$ between two globally minimal real analytic hypersurfaces in ${\u2102}^{n}$ ($n\ge 2$) is real analytic at every point...

The rigidity properties of the local invariants of real algebraic Cauchy-Riemann structures imposes upon holomorphic mappings some global rational properties (Poincaré 1907) or more generally algebraic ones (Webster 1977). Our principal goal will be to unify the classical or recent results in the subject, building on a study of the transcendence degree, to discuss also the usual assumption of minimality in the sense of Tumanov, in arbitrary dimension, without rank assumption and for holomorphic...

We show that a CR function of class ${C}^{k}$, 0 ≤ k < ∞, on a tube submanifold of ${\u2102}^{n}$ holomorphically extends to the convex hull of the submanifold. The extension and all its derivatives through order k are shown to have nontangential pointwise boundary values on the original tube submanifold. The ${C}^{k}$-norm of the extension is shown to be no bigger than the ${C}^{k}$-norm of the original CR function.

We survey results on unique determination of local $CR$-automorphisms of smooth $CR$-manifolds and of local biholomorphisms of real-analytic $CR$-submanifolds of complex spaces by their jets of finite order at a given point. Examples generalizing [28] are given showing that the required jet order may be arbitrarily high.