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# Holomorphic Approximation and Complex Geometry

#### Awarded: NOK 6.4 mill.

Source:

Project Manager:

Project Number:

209751

Application Type:

Project Period:

2011 - 2016

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Subject Fields:

Exposing of points.
In 2012 Diederich, Fornæss and Wold used techniques developed by Forstneric and Wold for embedding of Riemann surfaces to show a much stronger result: for any point on the boundary one can map the domain so that near this point the domain looks like a strictly convex domain in a geometric sense. The article Exposing points on the boundaries of domains of finite 1-type is now published in Journa of Geometric Analysis. The results were further used by Fornæss and Wold to estimate the so called squeezing function on a strictly pseudoconvex domain: this function quantifies how much the domain looks like a ball observed from a point. The estimate has interesting consequences for asymptotic behaviour of invariant metrics, with estimates previously unknown. The article An estimate for the squeezing function and applications to invariant metrics is accepted for publication in PROC. of the KSCV 10. Fusheng Deng (post doctor on the project), Fornæss and Wold have developed this further, and are writing up the article Exposing points in complex spaces.
Holomorphic mappings/embedding of Riemann surfaces
In 2012 Wold showed that it is possible to construct a so-called Fatou-Bieberbach domain that intersects a complex line in a disk. This solves a problem from an article by Rosay and Rudin that layed the ground for the Andersen-Lempert theory, which is a the core of this project. The article A Fatou-Bieberbach domain intersecting the plane in the unit disk is now published in Proceedings of the AMS.
In 2013 Arosio, Bracci and Wold used the Andersen-Lempert theory to show that one can solve so-called Loewner PDEs with values in euclidean spaces. Arosio and Bracci had earlier showed that one can find solutions in abstract complex manifolds. The article Solving the Loewner PDE in Complete Starlike domains of C^N is published in Advances in Math., and the article Embedding Univalent functions in filtering Loewner Chains in higher dimensions is accepted in Proceedings of the AMS.
In 2013 Andrist and Wold developed the theory of embeddings of Riemann surfaces to embed Riemann surfaces into manifolds with the so-called density property. The article Riemann surfaces in Stein manifolds with density properties is accepted by Ann. Inst. Fourier. This was developed further to higher dimensions by Andrist, Forstneric, Ritter (post doctor on the project) and Wold, and the article Proper holomorphic embeddings into Stein manifolds with the density property is accepted in J. dAnalyse Math.
In 2013 Andrist and Wold showed that the automorphism group G of a Stein manifolds X with the density property has a universal element F, provided that G has a generalised translation T. This means that the family {T^{-n}FT^n} is dense in G, and this generalises a classical result by Birkhoff. The article Free dense subgroups of holomorphic automorphisms is accepted in Math. Z.
In 2014 Ritter (postdoctor on the project) used the theory of embedding of Riemann surfaces to show that any Riemann surfaces can be embedded into (C^*)^2 in prescribed topology classes provided that one allows deformation of the complex structure. The article A soft Oka principle for proper holomorphic embeddings of open Riemann surfaces into (C^*)^2 is accepted for publication by Crelles Journal.
In 2014 Kutzschebauch and Wold finished a work on developing the Andersen-Lempert theory in the non-compact setting. The article Carleman approximation by holomorphic automorphisms is accepted for publication in Crelles Journal. Related to so called Carleman approximation, Magnusson and Wold wrote the article A characterisation of totally real Carleman sets an applications to products of totally real sets which is accepted for publication in Math. Scand.
Oka theory
A central question in the Oka-theory is whether the complement C^n\B to a closed ball in C^n is a so-called Oka manifold. In short the question is whether C^n\B admits many holomorphic maps from C^m into it for any m. An a priori
weaker question is whether C^n\setminus B is elliptic in the sense of Gromov.
In 2014 Andrist, Shcherbina and Wold answered that latter in the negative. The main point is that vector bundles and so called sprays extend across B. The article Hartogs Extension Theorem for Vector Bundles and Sprays is accepted by Arkiv för Matematik.
Forstneric and Ritter worked on the first problem, and showed that the answer is yes if m

Vi vil løse varierende problemer innen feltet flere komplekse variable samt studere mulige anvendelser. Problemene tilhører underfeltene approksimasjonsteori, iterasjonsproblemer og lamineringer. Vi vil ansette to postdoktorer i toårsstillinger for å bid ra til prosjektet, og vi kommer til å ansette en ph.d.-student.

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3.1BILL. NOKtotal funding in the programme period586PROJECTShave received funding in the programme period3SOURCEShave financed the programme

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