Egmont Porten

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; Sussmanns orbit theorem; local and global aspects of holomorphic extension of CR functions; Tumanovs solution of Bishops 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; Painleve 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.

The Hartogs Extension Theorem on (n-1)-Complete Complex Spaces

Abstract Performing local extension from pseudoconcave boundaries along Levi-Hartogs figures and building a Morse-theoretical frame for the global control of monodromy, we establish a version of the Hartogs extension theorem which is valid in singular complex spaces (and currently not available by means of techniques), namely: For every domain Ω of an (n – 1)-complete normal complex space of pure dimension n ≧ 2, and for every compact set K ⊂ Ω such that Ω\K is connected, holomorphic or meromorphic functions in Ω\K extend holomorphically or meromorphically to Ω. Assuming that X is reduced and globally irreducible, but not necessarily normal, and that the regular part [Ω\K]reg is connected, we also show that meromorphic functions on Ω\K extend meromorphically to Ω.

A MORSE-THEORETICAL PROOF OF THE HARTOGS EXTENSION THEOREM

AbstractOne hundered years ago exactly, in 1906, Hartogs published a celebrated extension phenomenon (birth of Several Complex Variables), whose global counterpart was understood later: Holomorphic functions in a connected neighborhood V(∂Ω) of a connected boundary ∂Ω ⋐ℂn ≥ 2) do extend holomorphically and uniquely to the domain ό. Martinelli, in the early 1940’s, and Ehrenpreis in 1961 obtained a rigorous proof, using a new multidimensional integral kernel or a short % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafyOaIyRbae% baaaa!376B!

Local envelopes on CR manifolds 1

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METRICALLY THIN SINGULARITIES OF INTEGRABLE CR FUNCTIONS

argument, but it remained unclear how to derive a proof using only analytic discs, as did Hurwitz (1897), Hartogs (1906), and E. E. Levi (1911) in some special, model cases. In fact, known attempts (e.g., Osgood, 1929, Brown, 1936) struggled for monodromy against multivaluations, but failed to get the general global theorem.Moreover, quite unexpectedly, in 1998, Fornœss exhibited a topologically strange (nonpseudoconvex) domain όF ⊂ ℂ2 that cannot befitted in by holomorphic discs, when one makes the additional requirement that discs must all lie entirely inside όF. However, one should point out that the standard, unrestricted disc method usually allows discs to go outside the domain (just think of Levi pseudoconcavity).Using the method of analytic discs for local extensional steps and Morse-theoretical tools for the global topological control of monodromy, we show that the Hartogs extension theorem can be established in such a way.

Holomorphic Extension from Weakly Pseudoconcave CR Manifolds

In the present paper, we answer two questions raised by Jarnicki and Pflug: First, we show by a counterexample that the Hartogs-Bochner theorem is no longer true for non-separated Riemann domains. Secondly, we generalize a structure theorem of Dloussky, which examines the extension of singularity sets contained in analytic hypersurfaces, to non-separated Riemann domains. Moreover, our method yields a new proof of Dlousskys original result.

Enveloppe d'holomorphie locale des variétés CR et élimination des singularités pour les fonctions CR intégrables

We study the problem whether CR functions on a sufficiently pseudoconcave CR manifold M extend locally across a hypersurface of M. The sharpness of the main result will be discussed by way of a counter-example.

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

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. Jöricke 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.