Continuous CR Function Research Articles

ASYMPTOTIC APPROXIMATIONS AND A BOREL-TYPE RESULT FOR CR FUNCTIONS

We show that for any smooth CR manifold which has a peak function (in a weak sense) at some point p, formal power series at p can be approximated asymptotically by continuous CR functions. Furthermore, if the peak function satisfies a certain growth property, the asymptotic approximation is actually smooth. This in fact allows to invert, in a Borel-type theorem, the natural map taking a smooth CR function to its formal Taylor series.

The Hartogs phenomenon on weakly pseudoconcave hypersurfaces

In 2002, Henkin and Michel proved a local Hartogs phenomenon for real analytic CR functions on real analytic weakly pseudoconcave CR manifolds. The aim of the present article is to remove the assumptions on real analyticity in the case of weakly pseudoconcave hypersurfaces \({M\subset\mathbb{C}^n}\) . If M is a graph of class \({\mathcal{C}^2}\) and n ≥ 3, a global theorem is proved for the extension of holomorphic germs along M. If the appearing domains have nicely shaped boundary, a Hartogs theorem even holds for continuous CR functions, where the difference to the case of holomorphic germs relies on the possible presence of lower-dimensional CR orbits. Levi flat hypersurfaces in \({\mathbb{C}^2}\) require a separate treatment. Here an affirmative answer is given to the question of Tomassini, whether 2-spheres bound 3-balls in M.

On the holomorphic extension of CR functions from non-generic CR submanifolds of ℂn

We give a holomorphic extension result for continuous CR functions on a non-generic CR submanifold N of ℂn to complex transversal wedges with edges containing N. We show that given any v∈ℂn∖(TpN+iTpN), there exists a wedge of direction v whose edge contains a neighborhood of p in N, such that any continuous CR function defined locally near p extends holomorphically to that wedge.

An extension result of CR functions by a general Schwarz reflection principle

It is known that a real analytic CR function f on a real analytic, generic submanifold M in ℂ N can be holomorphically extended. A stronger result on a finite type, real analytic, generic submanifold M is found in which we assume f a continuous CR function with real analytic imaginary part Im(f). The idea is contained in a general Schwarz reflection principle in one complex variable.

Separate holomorphic extension of CR functions

We consider a real analytic foliation of \({\mathbb{C}^n}\) by complex analytic manifolds of dimension m issued transversally from a CR generic submanifold \({M\subset\mathbb{C}^n}\) of codimension m. We prove that a continuous CR function f on M which has separate holomorphic extension along each leaf, is holomorphic. When the leaves are cartesian straight planes, separate holomorphic extension along suitable selections of these planes suffices and f turns out to be holomorphic in a neighbourhood of their union. If M is a hypersurface we can also specify the side of the extension, regardless the leaves are straight or not.

CR Runge Sets on Hypersurface Graphs

This work contains an improvement of earlier results of Boggess and Dwilewicz regarding global approximation of CR functions by entire functions in the case of hypersurface graphs. In this work, we show that if ω, an open subset of a real hypersurface in ℂn, can be graphed over a convex subset in ℝ2n−1, then ω is CR-Runge in the sense that continuous CR functions on ω can be approximated by entire functions on ℂn in the compact open topology of ω. Examples are presented to show that this approximation result does not hold for graphed CR submanifolds in higher codimension.

CR-manifolds of dimension 5: A Lie algebra approach

We study real-analytic Levi degenerate hypersurfaces M in complex manifolds of dimension 3, for which the CR-automorphism group Aut(M) is a real Lie group acting transitively on M. We provide large classes of examples for such M, compute the corresponding groups Aut(M) and determine the maximal subsets of M that cannot be separated by global continuous CR-functions. It turns out that all our examples, although partly arising in different contexts, are locally CR-equivalent to the tube over the future light cone in 3-dimensional space-time.

Totally real discs in non-pseudoconvex boundaries

LetD be a relatively compact domain inC2 with smooth connected boundary ∂D. A compact setK⊂∂D is called removable if any continuous CR function defined on ∂D/K admits a holomorphic extension toD. IfD is strictly pseudoconvex, a theorem of B. Joricke states that any compactK contained in a smooth totally real discS⊂∂D is removable. In the present article we show that this theorem is true without any assumption on pseudoconvexity.

On the CR-structure of compact group orbits associated with bounded symmetric domains

Let E be a complex vector space of finite dimension and let K ⊂ GL(E) be a compact connected subgroup. Then for fixed a ∈ E the orbit K := K(a) is a real-analytic submanifold of E that inherits various structures from E. For instance, choosing a K-invariant positive definite inner product (x|y) on E makes K a Riemannian manifold on which K acts transitively by isometries. On the other hand, K inherits from E a Cauchy–Riemann structure (CR-structure), that is given by the distribution of the maximal complex subspaces Hx K := Tx K ∩ iTx K of the real tangent spaces Tx K ⊂ E, x ∈ K , together with the complex structure on every Hx K (multiplication by i). The subspace Hx K is called the holomorphic tangent space to K at x (see [9] and [4] as general references for CR-manifolds). Of interest for the geometry of the orbit K = K(a) with respect to its CR-structure is the study of the CR-functions (or more generally CR-mappings) on K , i.e. of smooth functions f : K → C that satisfy the tangential Cauchy–Riemann differential equations in the sense that the restriction of the differential df to every holomorphic tangent space is complex linear. For instance, all holomorphic functions defined in an open neighbourhood of K ⊂ E give by restriction real-analytic CR-functions on K . Actually, we deal with the more general continuous CR-functions on K (which satisfy by definition the tangential CR-equations in the distribution sense, or equivalently, which are locally uniform limits of sequences of smooth CR-functions due to the approximation theorem of BaouendiTreves [6]). In this context it is of interest to determine the space of all

On the Hartogs–Bochner phenomenon for CR functions in $P_2(\mathbb {C})$

Let M be a compact, connected C^2-smooth and globally minimal hypersurface M in P_2(C) which divides the projective space into two connected parts U^{+} and U^{-}. We prove that there exists a side, U^- or U^+, such that every continuous CR function on $M$ extends holomorphically to this side. Our proof of this theorem is a simplification of a result originally due to F. Sarkis (nternat. J. Math. 10 (1999), no. 7, 897--915).

Deformation of CR-manifolds, minimal points and CR-manifolds with the microlocal analytic extension property

LetM be a smooth CR-manifold embedded into ℂ n . Letp be a point inM and letC be a small truncated cone inM (in suitable Euclidean coordinates onM) with vertexp which “symmetry axis” is a real vector in the complex tangent space. Then one can deformM into a smooth CR-manifoldM d letting fixed all points outsideC in such a way thatp is a minimal point ofM d . This result is used to give a new proof of the fact that wedge extendability of continuous CR-functions propagates along the CR-orbits of a CR-manifold. It allows also to prove the following natural result which was conjectured by Trepreau. LetM be a smooth generic CR-manifold in ℂ n . SupposeM consists of one single CR-orbit. Then each continuous CR-function onM is wedge extendable at any point ofM. Uniqueness theorems for continuous CR-functions are derived.

Characterization of removable sets in strongly pseudoconvex boundaries

Let M be a complex-analytic manifold(1) of complex dimension n>2 . Given an open domain D c c M , a proper closed subset K of the boundary bD of D is called removable in case b D K is CLsmooth and every continuous CRfunction f on b D K has a continuous extension F to D K which is holomorphic on D. A general account of the subject of removable sets is given in [19], where one can also find most of the related references. One main result in this area is the following characterization of removable sets in the two-dimensional case (see [19; II.10]):

THE INVERSE OF A CR-HOMEOMORPHISM IS CR

Let Ω, [Formula: see text] be domains and M ⊂ Ω, M′ ⊂ Ω′ be C2-smooth relatively closed real hyper-surfaces. The main result of the article is: Theorem. Let f: M → M′ be a homeomorphic CR-mapping. Then the inverse mapping f−1: M′ → M is also CR. As one of the main tools in the proof of this theorem the following removable singularity result for continuous CR-functions on real hypersurfaces is derived (it also might be useful for other applications): Theorem.Let [Formula: see text] be a family of closed complex hypersurfaces in M. Put [Formula: see text]. Suppose that h is a continuous function on M which is CR in a neighborhood of each point [Formula: see text] and such that the restriction h\Sα is holomorphic for all α ∈ A. Then h is CR on M.

Uniqueness Theorems for CR Functions

AbstractLet M be a CR manifold embedded in ℭs of arbitrary codimension. M is called generic if the complex hull of the tangent space in all points of M is the whole ℭs. M is minimal (in sense of Tumanov) in p ϵ M if there does not exist any CR submanifold of M passing through p with the same CR dimension as M but of smaller dimension. Let M be generic and minimal in some point p ϵ M and N be a generic submanifold of M passing through p. We prove that a continuous CR function on M vanishes identically in some neigbourhood of p if its restriction to N either vanishes in p faster then some function with non‐integrable logarithm or it vanishes on a subset of N of positive measure.