Construction of labyrinths in pseudoconvex domains
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Mathematische Zeitschrift
Construction of labyrinths in pseudoconvex domains Stéphane Charpentier1 · Łukasz Kosinski ´ 2 Received: 10 July 2019 / Accepted: 23 November 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We build in a given pseudoconvex (Runge) domain D of C N an O(D)-convex set , every connected component of which is a holomorphically contractible (convex) compact set, enjoying the property that any continuous path γ : [0, 1) → D with limr →1 γ (r ) ∈ ∂ D and omitting has infinite length. This solves a problem left open in a recent paper by Alarcón and Forstneriˇc. Keywords Several complex variables · Labyrinth Mathematics Subject Classification Primary 32H02; Secondary 32C22
1 Introduction Alarcón and Forstneriˇc recently proved that the Euclidean ball B N of C N , N > 1, admits a nonsingular holomorphic foliation by complete properly embedded holomorphic discs [1, Theorem 1]. They asked the natural question whether their result extends to any Runge pseudoconvex domains. As explained in [1, Remark 1], the main obstruction that appears is how to construct a suitable labyrinth in such a domain. Here and in the sequel we call labyrinth of a given pseudoconvex domain D in C N a set in D with the property that any continuous path γ : [0, 1) → D, with limr →∞ γ (r ) ∈ ∂ D, whose image does not intersect , has infinite length. Such sets were already built in pseudoconvex domains by Globevnik, by properly embedding the pseudoconvex domain as a submanifold of C2N +1 [7], thus reducing the problem to a construction in B N [6]. However Globevnik’s construction
Stéphane Charpentier partly supported by the Grant ANR-17-CE40-0021 of the French National Research Agency ANR (project Front). Łukasz Kosi´nski partly supported by the NCN Grant SONATA BIS no. 2017/26/E/ST1/00723.
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Stéphane Charpentier [email protected] Łukasz Kosi´nski [email protected]
1
Institut de Mathematiques, UMR 7373, Aix-Marseille Universite, 39 rue F. Joliot Curie, 13453 Marseille Cedex 13, France
2
Institute of Mathematics, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland
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S. Charpentier, Ł. Kosinski ´
in [6,7] did not provide with good topological properties of the connected components of the labyrinth, such as convexity or holomorphic contractibility. In [3] the authors simplified Globevnik’s construction building a labyrinth in B N whose connected components are balls in suitably chosen affine real hyperplanes. Alarcón and Forstneriˇc used a slight modification of this construction to obtain [1, Theorem 1]. The main aim of this short note is to overcome the difficulty pointed out in [1, Theorem 1] and extend the construction made in [3] to pseudoconvex domains. Theorem 1.1 Let D be a pseudoconvex domain in C N and let (Dn ) be a normal exhaustion of D by smooth strongly pseudoconvex domains that are O(D)-convex. Let also (Mn ) be a sequence of positive numbers. Then there are holomorphically contractible compact sets n ⊂ Dn+1 \D n such that D n ∪
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