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Egmont Porten

Publications and source records attributed to Egmont Porten.

18 recordsLinked to original sources
arXiv2024

Locally approximable CR functions, a sharp maximum modulus principle and holomorphic extension

We introduce a notion of locally approximable continuous CR functions on locally closed subsets of reduced complex spaces, generalizing both holomorphic functions and CR functions on CR submanifolds. Under additional assumptions of set-theoretical weak pseudoconcavity we prove optimal maximum modulus principles for these functions. Restricting to real submanifolds (possibly with CR singularities) of complexmanifolds, we generalize results on holomorphic extension known for CR submanifolds.

math.CV↗
arXiv2023

Truncated tube domains with multi-sheeted envelope

The present article is concerned with a group of problems raised by J. Noguchi and M. Jarnicki/P. Plug, namely whether the envelopes of holomorphy of truncated tube domains are always schlicht, i.e. subdomains of $\mathbb{C^n}$, and how to characterize schlichtness if this is not the case. By way of a counter-example homeomorphic to the 4-ball, we answer the first question in the negative. Moreover, it is possible that the envelopes have arbitrarily many sheets. The article is concluded by sufficient conditions for schlichtness in complex dimension two.

math.CV↗
arXiv2017

On the Hartogs extension theorem for unbounded domains in $\mathbb{C}^n$

Let $Ω\subset\mathbb{C}^n$, $n\geq 2$, be a domain with smooth connected boundary. If $Ω$ is relatively compact, the Hartogs-Bochner theorem ensures that every CR distribution on $\partialΩ$ has a holomorphic extension to $Ω$. For unbounded domains this extension property may fail, for example if $Ω$ contains a complex hypersurface. The main result in this paper tells that the extension property holds if and only if the envelope of holomorphy of $\mathbb{C}^n\backslash\overlineΩ$ is $\mathbb{C}^n$. It seems that it is a first result in the literature which gives a geometric characterization of unbounded domains in $\mathbb C^n$ for which the Hartogs phenomenon holds. Comparing this to earlier work by the first two authors and Z.~Słodkowski, one observes that the extension problem sensitively depends on a finer geometry of the contact of a complex hypersurface and the boundary of the domain.

math.CV↗
arXiv2017

A novel noncommutative KdV-type equation, its recursion operator, and solitons

A noncommutative KdV-type equation is introduced extending the Baecklund chart in [S. Carillo, M. Lo Schiavo, and C. Schiebold, SIGMA 12 (2016)]. This equation, called meta-mKdV here, is linked by Cole-Hopf transformations to the two noncommutative versions of the mKdV equations listed in [P.J. Olver and V.V. Sokolov Commun. Math. Phys. 193 (1998), Theorem 3.6]. For this meta-mKdV, and its mirror counterpart, recursion operators, hierarchies and an explicit solution class are derived.

math-ph↗
arXiv2012

On $\mathcal{C}^{\infty}$-hypoellipticity and extension of $CR$ functions

Let $M$ be a $CR$ submanifold of a complex manifold $X$. The main result of this article is to show that $CR$-hypoellipticity at $p_0\in{M}$ is necessary and sufficient for holomorphic extension of all germs of $CR$ functions to an ambient neighborhood in $X$. As an application, we obtain that $CR$-hypoellipticity implies the existence of generic embeddings and prove holomorphic extension for a large class of $CR$ manifolds satisfying a higher order Levi pseudoconcavity condition.

math.CV↗
arXiv2011

$\mathcal{C}^{\infty}$-hypoellipticity and extension of $CR$ functions

Let $M$ be a $CR$ submanifold of a complex manifold $X$. The main result of this article is to show that $CR$-hypoellipticity at $p_0\in{M}$ is necessary and sufficient for holomorphic extension of all germs of $CR$ functions to an ambient neighborhood in $X$. As an application, we obtain that $CR$-hypoellipticity implies the existence of generic embeddings and prove holomorphic extension for a large class of $CR$ manifolds satisfying a higher order Levi pseudoconcavity condition.

math.CV↗
arXiv2009

Complex vector fields and hypoelliptic partial differential operators

We prove a subelliptic estimate for systems of complex vector fields under some assumptions that generalize the essential pseudoconcavity for $CR$ manifolds and Hörmander's bracket condition for real vector fields. Applications are given to prove the hypoellipticity of first order systems and second order partial differential operators. Finally we describe a class of compact homogeneous CR manifolds for which the distribution of $(0,1)$ vector fields satisfies a subelliptic estimate. v2: minor revision, to appear in Ann. Inst. Fourier

math.AP↗
arXiv2007

The H-principle and Pseudoconcave CR Manifolds

The H-principle, which is the analogue, for CR manifolds, of the classical Hartogs principle in several complex variables, is known to be valid in the small on a pseudoconcave CR manifold of any codimension. However it fails in the large, as has been shown by the counterexample found in [HN1]. Hence there is an underlying obstruction to the global H-principle on a pseudoconcave CR manifold. The purpose of this note is to take the first steps toward a deeper understanding of this obstruction.

math.CV↗
arXiv2007

A geometrical proof of the Hartogs extension theorem

100 years ago exactly, in 1906, Hartogs published a celebrated extension phenomenon (birth of Several Complex Variables), whose global counterpart was stated in full generality later by Osgood (1929): holomorphic functions in a connected neighborhood V(bD) of a connected boundary bD contained in C^n (n >= 2) do extend holomorphically and uniquely to the domain D. It was a long-standing open problem to derive a proof using only analytic discs, as did Hurwitz (1897), Hartogs (1906) and E.E. Levi (1911) in some special, model cases. Quite unexpectedly, Fornaess in 1998 exhibited a topologically strange (nonpseudoconvex) domain D^F in C^2 that cannot be filled in by holomorphic discs, when one makes the additional requirement that discs must all lie entirely inside D^F. However, one should point out that the standard, unrestricted disc method usually allows discs to go outsise the domain (just think of Levi pseudoconcavity). Confirming these rather ancient expectations, we show that the global Hartogs extension theorem can be established in such a natural way.

math.CV↗
arXiv2007

The Hartogs extension theorem on (n-1)-complete complex spaces

Employing Morse theory for the global control of monodromy and the method of analytic discs for local extension, we establish a version of the global Hartogs extension theorem in a singular setting: for every domain D of an (n-1)-complete normal complex space X of pure dimension n >= 2 and for every compact set K in D such that D - K is connected, holomorphic or meromorphic functions in D - K extend holomorphically or meromorphically to D. Normality is an unvavoidable assumption for holomorphic extension, but we show that meromorphic extension holds on a reduced globally irreducible (not necessarily normal) X of pure dimension n >=2 provided that the regular part of D - K is connected.

math.CV↗
arXiv2007

Holomorphic extension of CR functions, envelopes of holomorphy and removable singularities

This is an extensive (published) survey on CR geometry, whose major themes are: formal analytic reflection principle; generic properties of Systems of (CR) vector fields; pairs of foliations and conjugate reflection identities; Sussmann's orbit theorem; local and global aspects of holomorphic extension of CR functions; Tumanov's solution of Bishop's equation in Hoelder classes with optimal loss of smoothness; wedge-extendability on C^2,a generic submanifolds of C^n consisting of a single CR orbit; propagation of CR extendability and edge-of-the-wedge theorem; Painlevé problem; metrically thin singularities of CR functions; geometrically removable singularities for solutions of the induced d-barre. Selected theorems are fully proved, while surveyed results are put in the right place in the architecture.

math.CV↗
arXiv2004

Wedge extendability of CR-meromorphic functions: the minimal case

In this article, we consider metrically thin singularities E of the solutions of the tangential Cauchy-Riemann operators on a C^{2,a}-smooth embedded Cauchy-Riemann generic manifold M (CR functions on M - E) and more generally, we consider holomorphic functions defined in wedgelike domains attached to M - E. Our main result establishes the wedge- and the L^1-removability of E under the hypothesis that the (\dim M-2)-dimensional Hausdorff volume of E is zero and that M and M\backslash E are globally minimal. As an application, we deduce that there exists a wedgelike domain attached to an everywhere locally minimal M to which every CR-meromorphic function on M extends meromorphically.

math.CV↗
arXiv2004

Characteristic foliations on maximally real submanifolds of C^n and envelopes of holomorphy

Let S be an arbitrary real surface, with or without boundary, contained in a hypersurface M of the complex euclidean space \C^2, with S and M of class C^{2, a}, where 0 < a < 1. If M is globally minimal, if S is totally real except at finitely many complex tangencies which are hyperbolic in the sense of E. Bishop and if the union of separatrices is a tree of curves without cycles, we show that every compact K of S is CR-, W- and L^p-removable (Theorem~1.3). We treat this seemingly global problem by means of purely local techniques, namely by means of families of small analytic discs partially attached to maximally real submanifolds of C^n and by means of a thorough study of the relative disposition of the characteristic foliation with respect to the track on M of a certain half-wedge attached to M. This localization procedure enables us to answer an open problem raised by B. Jöricke: under a certain nontransversality condition with respect to the characteristic foliation, we show that every closed subset C of a C^{2,a}-smooth maximally real submanifold M^1 of a (n-1)-codimensional generic C^{2,a}-smooth submanifold of \C^n is CR-, W- and L^p-removable (Theorem~1.2'). The known removability results in CR dimension at least two appear to be logical consequences of Theorem~1.2'. The main proof (65p.) is written directly in arbitrary codimension. Finally, we produce an example of a nonremovable 2-torus contained in a maximally real 3-dimensional maximally real submanifold, showing that the nontransversality condition is optimal for universal removability. Numerous figures are included to help readers who are not insiders of higher codimensional geometry.

math.CV↗
arXiv1999

Metrically thin singularities of integrable CR functions

In this article, we consider metrically thin singularities A of the tangential Cauchy-Riemann operator on smoothly embedded Cauchy-Riemann manifolds M. The main result states removability within the space of locally integrable functions on M under the hypothesis that the (dim M-2)-dimensional Hausdorff volume of A is zero and that the CR-orbits of M and M-A are comparable.

math.CV↗
arXiv1999

On the local meromorphic extension of CR meromorphic mappings

Let $M$ be a generic CR submanifold in $\C^{m+n}$, $m= CRdim M \geq 1$,$n=codim M \geq 1$, $d=dim M = 2m+n$. A CR meromorphic mapping (in the sense of Harvey-Lawson) is a triple $(f,{\cal D}_f, [Γ_f])$, where: 1. $f: {\cal D}_f \to Y$ is a ${\cal C}^1$-smooth mapping defined over a dense open subset ${\cal D}_f$ of $M$ with values in a projective manifold $Y$; 2. The closure $Γ_f$ of its graph in $\C^{m+n} \times Y$ defines a oriented scarred ${\cal C}^1$-smooth CR manifold of CR dimension $m$ (i.e. CR outside a closed thin set) and 3. Such that $d[Γ_f]=0$ in the sense of currents. We prove in this paper that $(f,{\cal D}_f, [Γ_f])$ extends meromorphically to a wedge attached to $M$ if $M$ is everywhere minimal and ${\cal C}^ω$ (real analytic) or if $M$ is a ${\cal C}^{2,α}$ globally minimal hypersurface.

math.CV↗