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Two-dimensional slices of nonpseudoconvex domains with rough boundary

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Abstract

For a domain \(D\subset {\mathbb C}^n,\; n\ge 3\), the set \(E\) is defined as the set of all points \(z\in {\mathbb C}^n\) for which the intersection of \(D\) with every complex \(2\)-plane through \(z\) is pseudoconvex. For \(D\) nonpseudoconvex, it is shown that \(E\) is contained in an affine subspace of codimension \(2\). This results solves a problem raised by Nikolov and Pflug.

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References

  1. Grauert, H., Remmert, R.: Konvexität in der komplexen analysis. Comment. Math. Helvetici 31, 152–183 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  2. Hitotumatu, S.: On some conjectures concerning pseudo-convex domains. J. Math. Soc. Jpn. 6, 177–195 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  3. Hörmander, L.: An Introduction to Complex Analysis in Several Variables. North-Holland, Amsterdam (1990)

    MATH  Google Scholar 

  4. Jarnicki, M., Pflug, P.: Extension of holomorphic functions. De Gruyter Expositions in Mathematics, vol. 34. Walter de Gruyter, Berlin (2000)

    Book  Google Scholar 

  5. Jacobson, R.: Pseudoconvexity is a two-dimensional phenomenon. Preprint arXiv:0907.1304v1

  6. Nikolov, N., Pflug, P.: Two-dimensional slices on non-pseudoconvex open sets. Math. Z. 272, 381–388 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Nikolov, N., Thomas, P.J.: Rigid characterizations of pseudoconvex domains. Indiana Univ. Math. J. 61, 1313–1323 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Porten, E.: On the Hartogs-phenomenon and extension of analytic hypersurfaces in non-separated Riemann domains. Complex Var. 47, 325–332 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The author would like to thank Nikolai Nikolov and Peter Pflug for valuable remarks. In particular, I am grateful to Peter Pflug for pointing out a confusion in the original proof of Lemma 2.2, and for providing an elegant argument, which both rectifies and shortens the original reasoning.

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  1. Department for Science Education and Mathematics, Mid Sweden University, Sundsvall , 85170, Sweden

    Egmont Porten

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  1. Egmont Porten
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Correspondence to Egmont Porten.

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Porten, E. Two-dimensional slices of nonpseudoconvex domains with rough boundary. Math. Z. 278, 19–23 (2014). https://doi.org/10.1007/s00209-014-1302-x

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  • DOI: https://doi.org/10.1007/s00209-014-1302-x

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