Mathematical Research Letters

Volume 27 (2020)

Number 6

Holomorphic families of Fatou–Bieberbach domains and applications to Oka manifolds

Pages: 1697 – 1706



Franc Forstnerič (Faculty of Mathematics and Physics, University of Ljubljana, Slovenia; and Department of Mathematics, Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia)

Erlend Fornæss Wold (Department of Mathematics, University of Oslo, Norway)


We construct holomorphically varying families of Fatou–Bieberbach domains with given centres in the complement of any compact polynomially convex subset $K$ of $\mathbb{C}^n$ for $n \gt 1$. This provides a simple proof of the recent result of $Y$. Kusakabe to the effect that the complement $\mathbb{C}^n \setminus K$ of any polynomially convex subset $K$ of $\mathbb{C}^n$ is an Oka manifold. The analogous result is obtained with $\mathbb{C}^n$ replaced by any Stein manifold with the density property.

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The first-named author is supported by the research program P1-0291 and grant J1-9104 from ARRS, Republic of Slovenia, and by the Stefan Bergman Prize 2019 from the American Mathematical Society. The second-named author is supported by the RCN grant 240569 from Norway.

Received 25 May 2020

Accepted 30 September 2020

Published 17 February 2021