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Generalized Hénon Mappings and Foliation by Injective Brody Curves

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We consider a finite composition of generalized Hénon mappings \({\mathfrak {f}}:{\mathbb {C}}^2\rightarrow {\mathbb {C}}^2\) and its Green function \({\mathfrak {g}}^+:{\mathbb {C}}^2\rightarrow {\mathbb {R}}_{\ge 0}\) (see Sect. 2). It is well known that each level set \(\{{\mathfrak {g}}^+=c\}\) for \(c>0\) is foliated by biholomorphic images of \({\mathbb {C}}\) and each leaf is dense. In this paper, we prove that each leaf is actually an injective Brody curve in \(\mathbb {P}^2\) (see Sect. 4). We also study the behavior of the level sets of \({\mathfrak {g}}^+\) near infinity.

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Acknowledgements

The author would like to thank Prof. John Erik Fornæss and Prof. Nessim Sibony for introducing this problem and for their advice on this topic. The author thanks the referee for careful reading and for suggestions. This work was supported in part by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIP) (No. 2011-0030044).

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  1. KIAS, 85 Hoegiro, Dongdaemun-gu, Seoul, 130-722, Republic of Korea

    Taeyong Ahn

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  1. Taeyong Ahn
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Ahn, T. Generalized Hénon Mappings and Foliation by Injective Brody Curves. J Geom Anal 28, 317–334 (2018). https://doi.org/10.1007/s12220-017-9821-4

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