Let $D\subset \subset {ℂ}^n,n\ge 2$, be a domain with $C^2$-boundary and $K\subset \partial D$ be a compact set such that $\partial D\setminus K$ is connected. We study univalent analytic extension of CR-functions from $\partial D\setminus K$ to parts of $D$. Call $K$ CR-convex if its $A\left(D\right)$-convex hull, $A\left(D\right)-\mathrm{hull}\left(K\right)$, satisfies $K=\partial D\cap A\left(D\right)-\mathrm{hull}\left(K\right)$ ($A\left(D\right)$ denoting the space of functions, which are holomorphic on $D$ and continuous up to $\partial D$). The main theorem of the paper gives analytic extension to $\partial D\setminus A\left(D\right)-\mathrm{hull}\left(K\right)$, if $K$ is CR- convex.

Let S ⊂ ℂⁿ, n ≥ 3, be a compact connected 2-codimensional submanifold having the following property: there exists a Levi-flat hypersurface whose boundary is S, possibly as a current. Our goal is to get examples of such S containing at least one special 1-hyperbolic point: a sphere with two horns, elementary models and their gluings. Some particular cases of S being a graph are also described.

We study effectively the Cartan geometry of Levi-nondegenerate C 6-smooth hypersurfaces M 3 in ℂ2. Notably, we present the so-called curvature function of a related Tanaka-type normal connection explicitly in terms of a graphing function for M, which is the initial, single available datum. Vanishing of this curvature function then characterizes explicitly the local biholomorphic equivalence of such M 3 ⊂ ℂ2 to the Heisenberg sphere ℍ3, such M’s being necessarily real analytic.

Let f : M → M' be a CR homeomorphism between two minimal, rigid polynomial varieties of Cn without holomorphic curves. We show that f extends biholomorphically in a neighborhood of M if f extends holomorphically in a neighborghood of a point p0 ∈ M or if f is of class C1. In the other hand, in case M and M' are two algebraic hypersurfaces, we obtain the extension without supplementary conditions.

We give a homotopy formula for the $\partial {̅}_b$ operator on a domain with (q+k)-concave boundary. As a consequence we show that the dimension of the $\partial {̅}_b$-cohomology groups for some CR manifolds q-concave at infinity is finite.

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

We present a large class of homogeneous 2-nondegenerate CR-manifolds $M$, both of hypersurface type and of arbitrarily high CR-codimension, with the following property: Every CR-equivalence between domains $U$, $V$ in $M$ extends to a global real-analytic CR-automorphism of $M$. We show that this class contains $G$-orbits in Hermitian symmetric spaces $Z$ of compact type, where $G$ is a real form of the complex Lie group $Aut{\left(Z\right)}^0$ and $G$ has an open orbit that is a bounded symmetric domain of tube type.

For large classes of complex Banach spaces (mainly operator spaces) we consider orbits of finite rank elements under the group of linear isometries. These are (in general) real-analytic submanifolds of infinite dimension but of finite CR-codimension. We compute the polynomial convex hull of such orbits $M$ explicitly and show as main result that every continuous CR-function on $M$ has a unique extension to the polynomial convex hull which is holomorphic in a certain sense. This generalizes to infinite...

Let M be a generic CR submanifold in ${ℂ}^{m+n}$, m = CR dim M ≥ 1, n = codim M ≥ 1, d = dim M = 2m + n. A CR meromorphic mapping (in the sense of Harvey-Lawson) is a triple $(f{,}_f,\left[{\Gamma }_f\right])$, where: 1) $f{:}_f\to Y$ is a ¹-smooth mapping defined over a dense open subset ${}_f$ of M with values in a projective manifold Y; 2) the closure ${\Gamma }_f$ of its graph in ${ℂ}^{m+n}\times Y$ defines an oriented scarred ¹-smooth CR manifold of CR dimension m (i.e. CR outside a closed thin set) and 3) $d\left[{\Gamma }_f\right]=0$ in the sense of currents. We prove that $(f{,}_f,\left[{\Gamma }_f\right])$ extends meromorphically to a wedge...

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 study Levi-flat real analytic hypersurfaces with singularities. We prove that the Levi foliation on the regular part of the hypersurface can be holomorphically extended, in a suitable sense, to neighbourhoods of singular points.

We study the regularity problem for Cauchy Riemann maps between hypersurfaces in Cn. We prove that a continuous Cauchy Riemann map between two smooth C∞ pseudoconvex decoupled hypersurfaces of finite D'Angelo type is of class C∞.