Abstracts

John Erik Fornæss (U of M), Short ℂ²

We construct domains in ℂ² which have some characteristic features of ℂ², but who nevertheless are not biholomorphic to ℂ².

Sönmez Şahutoğlu (U of M), A hull with no analytic structure (after Stolzenberg)

We will read a paper by Stolzenberg. In this paper the author constructs a compact set K in two dimensional complex space such that the polynomial hull of K does not contain an analytic disc.

Chris Hammond (U of M), On the Topological Relationship between a Domain and its Envelope of Holomorphy

We will look at a paper of Hans Kerner relating the fundamental group of a domain to the fundamental group of its envelope. We will also look at a paper of Fornæss and Zame which suggests that no better results are possible. Finally, we will make some observations about this relationship in ℂ².

Jean Ruppenthal (U of Bonn), The ∂-equation on singular Stein spaces

One method to attack the ∂-problem on singular Stein spaces is to use Hironaka's desingularization in order to blow up the singularities, and to investigate the ∂-equation on the resulting regular complex manifold. We will talk about problems and prospects of that approach. Particularly, it seems to be quite promising in the case of L^∞-estimates.

Andrei Iordan (Institut de Mathématiques de Jussieu), Complex tangential curves of constant curvature in the unit ball of ℂ² and homogeneous polynomials

We prove that a non-constant homogeneous polynomial P on ℂ² having the property that P=1 on a complex-tangential real analytic curve of the boundary of the unit ball reduces to a monomial by a unitary change of variables. This is a positive answer to conjectures of H.O. Kim.

Lina Lee (U of M), Estimation of the Kobayashi metric in the normal direction

We present recent work on the asymptotic behavior of the Kobayashi metric. We discuss whether the Kobayashi metric blows up at a rate comparable to one over distance to the boundary in the normal direction if the domain is pseudoconvex.

Liz Vivas (U of M), A long ℂ² which is not Stein

We will look at recent work of Erlend Wold. In this paper the author answers a question posed by Fornæss and Sibony: Is any long ℂ² biholomorphic to ℂ²? A complex manifold is a long ℂ² if it is a union of proper subsets which are biholomorphic to ℂ². The answer is negative and the author constructs a long ℂ² which is not Stein.

Elizabeth Wulcan (Chalmers University), Hypermeromorphic currents

(Joint work with Mats Andersson) We introduce a class of currents, which we call hypermeromorphic and which includes meromorphic currents such as principal value and residue currents. We discuss some basic properties; in particular, we show that hypermeromorphic currents allow for multiplication with characteristic functions 1_W, where W is a variety or more generally a constructible set. The motivation is that we by such a calculus obtain a natural geometric decomposition of residue currents.

Joël Merker (ENS, Paris), Local CR geometry, CR orbits, Hilbert transform in Hölder spaces and the local Bishop equation

The talk will be designed for non-experts. CR manifolds carry a constant rank induced tangential antiholomorphic distribution which, generically, is not Frobenius-integrable. Beyond Chow's or Nagano's Lie bracket spanning condition, Sussmann orbits are the adequate concept in the smooth category. Usually, holomorphic discs attached to CR manifolds are constructed in Hölder spaces, because of their flexibility for norm computations, Picard iteration, and implicit function theorem.

Egmont Porten (Mid Sweden U), Geometry of CR orbits and holomorphic extension

The talk will be for a general audience. One of the fundamental questions in the study of CR functions on embedded CR manifolds is whether they extend holomorphically to open wedges in the ambient space. If the answer is positive, one can deduce very strong structural information. We will explain the optimal extension theorems which link holomorphic extension with the structure of CR orbits. The original proofs are due to Trépreau and Tumanov in the local, and to Jöricke and Merker in the global case. We will explain a recent, relatively simple argument which shows a very transparent link between differential geometry and extension properties and does apply to the local and global settings uniformly.

Egmont Porten (Mid Sweden U), Levi Flat Fillings of Spheres and the Continuity Principle

The classical continuity principle tells how to use families of holomorphic discs in the geometric study of envelopes of holomorphy. In order to gain flexibility in applications, it is desirable to have more general versions of the continuity principle applying to families of general holomorphic curves. In the talk, we will mainly concentrate on an example illustrating various hidden obstructions to possible generalizations. We will construct a 2-sphere S⊂ℂ² bounding an embedded Levi flat 3-ball along which holomorphic extension from neighborhoods of S fails. Interestingly, the same S also bounds an immersed Levi flat ball with good extension properties. This is joint work with Burglind Jöricke, Bonn.