Extension Of CR Functions Research Articles

Codimension Two CR Singular Submanifolds and Extensions of CR Functions

Let $$M \subset {\mathbb {C}}^{n+1}$$ , $$n \ge 2$$ , be a real codimension two CR singular real analytic submanifold that is nondegenerate and holomorphically flat. We prove that every real analytic function on M that is CR outside the CR singularities extends to a holomorphic function in a neighbourhood of M. Our motivation is to prove the following analogue of the Hartogs–Bochner theorem. Let $$\Omega \subset {\mathbb {C}}^n \times {\mathbb {R}}$$ , $$n \ge 2$$ , be a bounded domain with a connected real analytic boundary such that $$\partial \Omega $$ has only nondegenerate CR singularities. We prove that if $$f :\partial \Omega \rightarrow {\mathbb {C}}$$ is a real analytic function that is CR at CR points of $$\partial \Omega $$ , then f extends to a holomorphic function on a neighbourhood of $${\overline{\Omega }}$$ in $${\mathbb {C}}^n \times {\mathbb {C}}$$ .

Extension of CR functions from boundaries in Cn x R

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 function that is CR at CR points of $\partial \Omega$ (when $n=1$ we require separate holomorphic extensions for each real parameter). Then $f$ extends to a smooth CR function on $\bar{\Omega}$, that is, an analogue of Hartogs-Bochner holds. In addition, if $f$ and $\partial \Omega$ are real-analytic, then $f$ is the restriction of a function that is holomorphic on a neighborhood of $\bar{\Omega}$ in ${\mathbb C}^{n+1}$. An immediate application is a (possibly singular) solution of the Levi-flat Plateau problem for codimension 2 submanifolds that are CR images of $\partial \Omega$ as above. The extension also holds locally near nondegenerate, holomorphically flat, elliptic CR singularities.

An integral formula in Kähler geometry with applications

We establish an integral formula on a smooth, precompact domain in a Kähler manifold. We apply this formula to study holomorphic extension of CR functions. Using this formula, we also prove some geometric inequalities when the boundary has positive Hermitian mean curvature.

Approximation of Singular Discs for CR Extension

We introduce a class of singular discs, modelled on the one introduced by Zaitsev, Zampieri and the author. We show that they gain regularity from reparametrization and from taking powers. We set up a Bishop equation to attach these discs to a CR manifold with smooth dependence on parameters. By these tools we get extension of CR functions from wedges endowed with the sector property.

Wermer type sets and extension of CR functions

For each $n\geq2$ we construct an unbounded closed pseudoconcave complete pluripolar set $\mathcal E$ in $\mathbb C^n$ which contains no analytic variety of positive dimension (we call it a \textit{Wermer type set}). We also construct an unbounded strictly pseudoconvex domain $\Omega$ in $\mathbb C^n$ and a smooth $CR$ function $f$ on $\partial\Omega$ which has a single-valued holomorphic extension exactly to the set $\bar\Omega\setminus\mathcal E$.}

A Remark on Extensions of CR Functions from Hyperplanes

AbstractIn the characterization of the range of the Radon transform, one encounters the problem of the holomorphic extension of functions defined on ℝ2 \ Δℝ (where Δℝ is the diagonal in ℝ2) and which extend as “separately holomorphic” functions of their two arguments. In particular, these functions extend in fact to ℂ2 \ Δℂ where Δℂ is the complexification of Δℝ. We take this theorem from the integral geometry and put it in the more natural context of the CR geometry where it accepts an easier proof and amore general statement. In this new setting it becomes a variant of the celebrated “edge of the wedge” theorem of Ajrapetyan and Henkin.

Rays condition and extension of CR functions from manifolds of higher type

We prove that CR functions defined in a wedge inside a CR manifold extend to be CR (or holomorphic) in the directions given by the higher order generalization of the Levi form taken at complex tangent vectors satisfying the so-called rays condition. This generalizes extension results by Boggess-Polking [7], Baouendi-Treves [3], Fornaess-Rea [10] and the second and the third authors [18] and puts them into a unified frame.

Propagation of CR extendibility along complex tangent directions

We discuss the propagation along CR curves of extendibility of CR functions in the framework of the theory of partial lifts of CR curves and of analytic discs introduced by the authors in (Baracco, L. and Zampieri, G., 2001, Analytic discs and extension of CR functions. Compositio Mathematica, 127, 289–295) and (Baracco, L. and Zampieri, G., 2002, Tangent discs and extension of CR functions to wedges of . J. Geom. Analysis, 12(1), 1–7).

Extension of CR-functions on wedges

We generalize the extension theorem of Ajrapetyan-Henkin [AH81] and Boggess-Polking [BP82] to wedge-like domains V in a generic manifold M. For each complex tangent vector to M, we introduce its complex angle with respect to V and show that CR-functions on V have natural extension in the direction of the Levi form of any tangent vector whose angle is greater than π/2. This gives an explanation of the extension phenomenon recently discovered by Eastwood-Graham [EG99,EG01]. We further demonstrate by an example that the angle π/2 is optimal and cannot be replaced by any smaller angle. We conclude by giving applications to holomorphic extension to a full neighborhood of the reference point.

Analytic Discs and Extension of CR Functions

Let M be a manifold of X = n, A a small analytic disc attached to M, zo a point of ∂A where A is tangent to M, z1 another point of ∂A where M extends to a germ of manifold M1 with boundary M. We prove that CR functions on M which extend to M1 at z1 also extend at zo to a new manifold M2. The directions M1 and M2 point to, are related by a sort of connection associated to A which is dual to the connection obtained by attaching ‘partial analytic lifts’ of A to the co-normal bundle to M in X.

Subaveraging estimates for CR functions on a hypersurface of C n

For a real smooth hypersurface M of C n, we give a subaveraging estimate in some points of M for CR functions defined near these points. We obtain this by establishing subesti-mates at some point which lies outside of M and whose normal projection on M is a point of finite type. Secondly, we study the equivalence between the extension of CR functions and the existence of analytic discs.

Spinning analytic discs and domains of dependence

In this paper we introduce a new technique of spinning analytic discs, which allows us to control the direction of CR extension of CR functions from rigid CR manifolds.

On extension of CR functions from piecewise smooth manifolds into a wedge

On extension of CR functions from piecewise smooth manifolds into a wedge

About the holomorphic extension of CR functions on real hypersurfaces in complex space

About the holomorphic extension of CR functions on real hypersurfaces in complex space

Microlocal hypo-analyticity and extension of CR functions

Microlocal hypo-analyticity and extension of CR functions

The extension of CR functions to one side of a submanifold of ${\bf C}^n$.

The extension of CR functions to one side of a submanifold of ${\bf C}^n$.

Extensions of CR-Functions

Extensions of CR-Functions