CR Singularities Research Articles

Stability of the hull(s) of an n-sphere in [formula omitted

We study the (global) Bishop problem for small perturbations of Sn — the unit sphere of C×Rn−1 — in Cn. We show that if S⊂Cn is a sufficiently-small perturbation of Sn (in the C3-norm), then S bounds an (n+1)-dimensional ball M⊂Cn that is foliated by analytic disks attached to S. Furthermore, if S is either smooth or real analytic, then so is M (up to its boundary). Finally, if S is real analytic (and satisfies a mild condition), then M is both the envelope of holomorphy and the polynomially convex hull of S. This generalizes the previously known case of n=2 (CR singularities are isolated) to higher dimensions (CR singularities are nonisolated).

On CR singular CR images

We say that a CR singular submanifold [Formula: see text] has a removable CR singularity if the CR structure at the CR points of [Formula: see text] extends through the singularity as an abstract CR structure on [Formula: see text]. We study such real-analytic submanifolds, in which case removability is equivalent to [Formula: see text] being the image of a generic real-analytic submanifold [Formula: see text] under a holomorphic map that is a diffeomorphism of [Formula: see text] onto [Formula: see text], what we call a CR image. We study the stability of the CR singularity under perturbation, the associated quadratic invariants, and conditions for removability of a CR singularity. A lemma is also proved about perturbing away the zeros of holomorphic functions on CR submanifolds, which could be of independent interest.

On the Levi-flat Plateau problem

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 diffeomorphic image via a real-analytic CR map of a real-analytic hypersurface in ${\mathbb C}^n \times {\mathbb R}$ with only nondegenerate CR singularities. Then there exists a unique compact real-analytic Levi-flat hypersurface, nonsingular except possibly for self-intersections, with boundary $M$. We also study boundary regularity of CR automorphisms of domains in ${\mathbb C}^n \times {\mathbb R}$.

Real submanifolds of maximum complex tangent space at a CR singular point, II

We study a germ of real analytic n-dimensional submanifold of $C^n$ that has a complex tangent space of maximal dimension at a CR singularity. Under the condition that its complexification admits the maximum number of deck transformations, we first classify holomorphically its quadratic CR singularity. We then study its transformation to a normal form under the action of local (possibly formal) biholomorphisms at the singularity. We first conjugate formally its associated reversible map $\sigma$ to suitable normal forms and show that all these normal forms can be divergent. We then construct a unique formal normal form under a non degeneracy condition.

Geometric and analytic problems for a real submanifold in ℂ n with CR singularities

In this survey article, we present some known results and also propose some open questions related to the analytic and geometric aspects of Bishop submanifolds in a complex space. We mainly focus on those problems that the author and his coauthors have recently worked on. The article also contains an example of a Bishop submanifold in ℂ3 of real codimension two, which cannot be quadratically flattened at a CR singular point but is CR non-minimal at any CR point. This provides a counter-example to a question asked in a private communication by Zaistev (2013).

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.

Normal forms and degenerate CR singularities

Let (z, w) be the coordinates in . We construct a normal form for a class of real formal surfaces defined near a degenerate CR singularity as followswhere is a real-valued homogeneous polynomial in of degree such that the coefficients of and are vanishing.

Real submanifolds of maximum complex tangent space at a CR singular point, I

We study a germ of real analytic n-dimensional submanifold of C n that has a complex tangent space of maximal dimension at a CR singularity. Under some assumptions , we show its equivalence to a normal form under a local biholomorphism at the singularity. We also show that if a real submanifold is formally equivalent to a quadric, it is actually holomorphically equivalent to it, if a small divisors condition is satisfied. Finally, we investigate the existence of a complex submanifold of positive dimension in C n that intersects a real submanifold along two totally and real analytic submanifolds that intersect transversally at a possibly non-isolated CR singularity.

A normal form for a real 2-codimensional submanifold in [formula omitted] near a CR singularity

We construct a formal normal form for a class of real submanifolds M⊂CN+1 of codimension 2 near a CR singularity approximating the sphere. Our result gives a generalization of Huang–Yin’s normal form in C2 to the higher dimensional analog case.

The local polynomial hull near a degenerate CR singularity: Bishop discs revisited

Let $${\mathcal{S}}$$ be a smooth real surface in $${\mathbb{C}^2}$$ and let $${p\in\mathcal{S}}$$ be a point at which the tangent plane is a complex line. How does one determine whether or not $${\mathcal{S}}$$ is locally polynomially convex at such a p—i.e. at a CR singularity? Even when the order of contact of $${T_p(\mathcal{S})}$$ with $${\mathcal{S}}$$ at p equals 2, no clean characterisation exists; difficulties are posed by parabolic points. Hence, we study non-parabolic CR singularities. We show that the presence or absence of Bishop discs around certain non-parabolic CR singularities is completely determined by a Maslov-type index. This result subsumes all known facts about Bishop discs around order-two, non-parabolic CR singularities. Sufficient conditions for Bishop discs have earlier been investigated at CR singularities having high order of contact with $${T_p(\mathcal{S})}$$ . These results relied upon a subharmonicity condition, which fails in many simple cases. Hence, we look beyond potential theory and refine certain ideas going back to Bishop.

Levi-flat hypersurfaces with real analytic boundary

Let $X$ be a Stein manifold of dimension at least 3. Given a compact codimension 2 real analytic submanifold $M$ of $X$, that is the boundary of a compact Levi-flat hypersurface $H$, we study the regularity of $H$. Suppose that the CR singularities of $M$ are an $\mathcal{O}(X)$-convex set. For example, suppose $M$ has only finitely many CR singularities, which is a generic condition. Then $H$ must in fact be a real analytic submanifold. If $M$ is real algebraic, it follows that $H$ is real algebraic and in fact extends past $M$, even near CR singularities. To prove these results we provide two variations on a theorem of Malgrange, that a smooth submanifold contained in a real analytic subvariety of the same dimension is itself real analytic. We prove a similar theorem for submanifolds with boundary, and another one for subanalytic sets.

Unfolding CR Singularities

A notion of unfolding, or multi-parameter deformation, of CR singularities of real submanifolds in complex manifolds is proposed, along with a definition of equivalence of unfoldings under the action of a group of analytic transformations. In the case of real surfaces in complex $2$-space, deformations of elliptic, hyperbolic, and parabolic points are analyzed by putting the parameter-dependent real analytic defining equations into normal forms up to some order. For some real analytic unfoldings in higher codimension, the method of rapid convergence is used to establish real algebraic normal forms. Table of Contents: Introduction; Topological considerations; Local defining equations and transformations; A complexification construction; Real surfaces in $\mathbb{C}^2$; Real $m$-submanifolds in $\mathbb{C}^n, m < n$; Rapid convergence proof of the main theorem; Some other directions; Bibliography. (MEMO/205/962)

CR singularities of real fourfolds in $\mathbb{C}^3$

CR singularities of real 4-submanifolds in complex 3-space are classified by using local holomorphic coordinate changes to transform the quadratic coefficients of the real analytic defining equation into a normal form. The quadratic coefficients determine an intersection index, which appears in global enumerative formulas for CR singularities of compact submanifolds.

Real equivalence of complex matrix pencils and complex projections of real Segre varieties

Quadratically parametrized maps from a product of real projective spaces to a com- plex projective space are constructed as the composition of the Segre embedding with a projection. A classification theorem relates equivalence classes of projections to equivalence classes of complex matrix pencils. One low-dimensional case is a family of maps whose images are ruled surfaces in the complex projective plane, some of which exhibit hyperbolic CR singularities. Another case is a set of maps whose images in complex projective 4-space are projections of the real Segre threefold.

Analytic stability of the CR cross-cap

For m < n, any real analytic m-submanifold of complex n-space with a nondegenerate CR singularity is shown to be locally equivalent, under a holomorphic coordinate change, to a fixed real algebraic variety defined by linear and quadratic polynomials. The situation is analogous to Whitney?s stability theorem for cross-cap singularities of smooth maps. The complex analyticity of the normalizing transformation is proved using a rapid convergence argument.

Polynomial approximation, local polynomial convexity, and degenerate CR singularities

We begin with the following question: given a closed disc D ¯ ⋐ C and a complex-valued function F ∈ C ( D ¯ ) , is the uniform algebra on D ¯ generated by z and F equal to C ( D ¯ ) ? When F ∈ C 1 ( D ) , this question is complicated by the presence of points in the surface S : = graph D ¯ ( F ) that have complex tangents. Such points are called CR singularities. Let p ∈ S be a CR singularity at which the order of contact of the tangent plane with S is greater than 2; i.e. a degenerate CR singularity. We provide sufficient conditions for S to be locally polynomially convex at the degenerate singularity p. This is useful because it is essential to know whether S is locally polynomially convex at a CR singularity in order to answer the initial question. To this end, we also present a general theorem on the uniform algebra generated by z and F, which we use in our investigations. This result may be of independent interest because it is applicable even to non-smooth, complex-valued F.

CR singularities of real threefolds in ℂ4

Abstract CR singularities of real threefolds in ℂ4 are classified by using holomorphic coordinate changes to transform the quadratic part of the real defining equations into one of a list of normal forms. In the non-degenerate case, it is shown that a real analytic manifold near a CR singular point is formally equivalent to a real algebraic model. Some degenerate cases also have this property.

Surfaces with degenerate CR singularities that are locally polynomially convex

Surfaces with degenerate CR singularities that are locally polynomially convex

Real congruence of complex matrix pencils and complex projections of real Veronese varieties

Quadratically parametrized maps from a real projective space to a complex projective space are constructed as projections of the Veronese embedding. A classification theorem relates equivalence classes of projections to real congruence classes of complex symmetric matrix pencils. The images of some low-dimensional cases include certain quartic curves in the Riemann sphere, models of the real projective plane in complex projective 4-space, and some normal form varieties for real submanifolds of complex space with CR singularities.