相关论文: Codimension two CR singular submanifolds and exten…

Let $\Omega \subset {\mathbb C}^n \times {\mathbb R}$ be a bounded domain with smooth boundary such that $\partial \Omega$ has only nondegenerate elliptic CR singularities, and let $f \colon \partial \Omega \to {\mathbb C}$ be a smooth…

Real-analytic CR functions on real-analytic CR singular submanifolds are not in general restrictions of holomorphic functions, unlike in the CR nonsingular case. We give a simple condition that completely characterizes those quadric CR…

We prove that if $M$ and $M'$ are algebraic hypersurfaces in $ C^ N$, i.e. both defined by the vanishing of real polynomials, then any sufficiently smooth CR mapping with Jacobian not identically zero extends holomorphically provided the…

Let M be a smooth locally embeddable CR manifold, having some CR dimension m and some CR codimension d. We find an improved local geometric condition on M which guarantees, at a point p on M, that germs of CR distributions are smooth…

Let $M$ be a connected real-analytic hypersurface in $\C^N$ and $\S$ the unit real sphere in $\C^{N'}$, $N'> N\geq 2$. Assume that $M$ does not contain any complex-analytic hypersurface of $\C^N$ and that there exists at least one strongly…

We prove that generic homologically nontrivial $(2n-1)$-parameter family of analytic discs attached by their boundaries to a CR manifold $\Omega$ in $\mathbb C^n, n \le 2$ tests CR functions: if a smooth function on $\Omega$ extends…

We establish the holomorphic wedge extendability of CR functions, defined on an everywhere locally minimal generic submanifold M of C^n and having singularities contained in a submanifold N of codimension 1, 2 or 3, assuming some…

In any positive CR-dimension and CR-codimension we provide a construction of real-analytic holomorphically nondegenerate CR-submanifolds, which are $C^\infty$ CR-equivalent, but are inequivalent holomorphically. As a corollary, we provide…

A general class of singular real hypersurfaces, called subanalytic, is defined. For a subanalytic hypersurface M in C^n, Cauchy-Riemann (or simply CR) functions on M are defined, and certain properties of CR functions discussed. In…

Let $\Omega\subset\mathbb{C}^n$, $n\geq 2$, be a domain with smooth connected boundary. If $\Omega$ is relatively compact, the Hartogs-Bochner theorem ensures that every CR distribution on $\partial\Omega$ has a holomorphic extension to…

A local uniqueness property of holomorphic functions on real-analytic nowhere minimal CR submanifolds of higher codimension is investigated. A sufficient condition called almost minimality is given and studied. A weaker necessary condition,…

This paper continues the previous studies in two papers of Huang-Yin [HY3-4] on the flattening problem of a CR singular point of real codimension two sitting in a submanifold in ${\mathbb C}^{n+1}$ with $n+1\ge 3$, whose CR points are…

The purpose of this paper is to organize some results on the local geometry of CR singular real-analytic manifolds that are images of CR manifolds via a CR map that is a diffeomorphism onto its image. We find a necessary (sufficient in…

Let $M$ be a smooth compact $CR$ manifold of $CR$ dimension $n$ and $CR$ codimension $k$, which has a certain local extension property $E$. In particular, if $M$ is pseudoconcave, it has property $E$. Then the field $\Cal K(M)$ of $CR$…

We say that a CR singular submanifold $M$ has a removable CR singularity if the CR structure at the CR points of $M$ extends through the singularity as an abstract CR structure on $M$. We study such real-analytic submanifolds, in which case…

In $\C^2=\R^2+i\R^2$ with coordinates $z=(z_1,z_2), z=x+iy$, we consider a function $f$ continuous on a domain $\Omega$ of $\R^2$ separately real analytic in $x_1$ and CR extendible to $y_2$ (resp. CR extendible to $y_2>0$). This means that…

It is proved that CR functions on a quadratic cone M in $\C^n$, n>1, admit one-sided holomorphic extension if and only if M does not have two-sided support, a geometric condition on M which generalizes minimality in the sense of Tumanov. A…

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…

We show that an arc-analytic subanalytic function on a complex manifold M, which is holomorphic near one point, is a holomorphic function on M. More generally, an arc-analytic subanalytic function on a real analytic CR-manifold M, which is…

We solve the Levi-flat Plateau problem in the following case. Let $M \subset {\mathbb C}^{n+1}$, $n \geq 2$, be a connected compact real-analytic codimension-two submanifold with only nondegenerate CR singularities. Suppose $M$ is a…