Abstract
We build in a given pseudoconvex (Runge) domain D of \({\mathbb {C}}^N\) an \(\mathcal O(D)\)-convex set \(\Gamma \), every connected component of which is a holomorphically contractible (convex) compact set, enjoying the property that any continuous path \(\gamma :[0,1)\rightarrow D\) with \(\lim _{r\rightarrow 1}\gamma (r)\in \partial D\) and omitting \(\Gamma \) has infinite length. This solves a problem left open in a recent paper by Alarcón and Forstnerič.
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References
Alarcón, A., Forstnerič, F.: A foliation of the ball by complete holomorphic discs. Math. Z. (2019). https://doi.org/10.1007/s00209-019-02430-6
Alarcón, A., Globevnik, J.: Complete embedded complex curves in the ball of \(\mathbb{C}^2\) can have any topology. Anal. PDE 751, 1987–1999 (2017)
Alarcón, A., Globevnik, J., Lòpez, F.J.: A construction of complete complex hypersurfaces in the ball with control on the topology. J. Reine Angew. Math. 751, 289–308 (2019)
Diederich, K., Fornæss, J.E., Wold, E.F.: Exposing points on the boundary of a strictly pseudoconvex or a locally convexifiable domain of finite 1-type. J. Geom. Anal. 24(4), 2124–2134 (2014)
Forstnerič, F.: Stein manifolds and holomorphic mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 56. Springer (2017)
Globevnik, J.: A complete complex hypersurface in the ball of \(\mathbb{C}^N\). Ann. Math. (2) 182(3), 1067–1091 (2015)
Globevnik, J.: Holomorphic functions unbounded on curves of finite length. Math. Ann. 364(3–4), 1343–1359 (2016)
Stout, E.: Polynomial Convexity, Progress in Mathematics, vol. 261. Birkhäuser Boston Inc, Boston (2007)
Yang, P.: Curvature of complex submanifolds in \({\mathbb{C}}^n\). In: Proc. Symp. Pure. Math. vol. 30, part 2, pp. 135–137. Amer. Math. Soc., Providence (1977)
Yang, P.: Curvature of complex submanifolds in \({\mathbb{C}}^n\). J. Differ. Geom. 12, 499–511 (1977)
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Stéphane Charpentier partly supported by the Grant ANR-17-CE40-0021 of the French National Research Agency ANR (project Front). Łukasz Kosiński partly supported by the NCN Grant SONATA BIS no. 2017/26/E/ST1/00723.
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Charpentier, S., Kosiński, Ł. Construction of labyrinths in pseudoconvex domains. Math. Z. 296, 1021–1025 (2020). https://doi.org/10.1007/s00209-020-02468-x
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DOI: https://doi.org/10.1007/s00209-020-02468-x