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Holomorphic functions unbounded on curves of finite length

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Abstract

Given a pseudoconvex domain , we prove that there is a holomorphic function f on D such that the lengths of paths \(p:\ [0,1] \rightarrow D\) along which \(\mathfrak {R}f\) is bounded above, with p(0) fixed, grow arbitrarily fast as \(p(1)\rightarrow bD\). A consequence is the existence of a complete closed complex hypersurface \( M\subset D\) such that the lengths of paths \(p:\ [0,1]\rightarrow M\), with p(0) fixed, grow arbitrarily fast as \(p(1)\rightarrow bD\).

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Acknowledgments

This work was supported by the Research Program P1-0291 from ARRS, Republic of Slovenia.

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  1. Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia

    Josip Globevnik

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  1. Josip Globevnik
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Correspondence to Josip Globevnik.

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Globevnik, J. Holomorphic functions unbounded on curves of finite length. Math. Ann. 364, 1343–1359 (2016). https://doi.org/10.1007/s00208-015-1253-5

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  • DOI: https://doi.org/10.1007/s00208-015-1253-5

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