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The Diederich–Fornæss Exponent and Non-existence of Stein Domains with Levi-Flat Boundaries

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Abstract

We study the Diederich–Fornæss exponent and relate it to non-existence of Stein domains with Levi-flat boundaries in complex manifolds. In particular, we prove that if the Diederich–Fornæss exponent of a smooth bounded Stein domain in an \(n\)-dimensional complex manifold is greater than \(k/n\), then it has a boundary point at which the Levi-form has rank greater than or equal to k.

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Notes

  1. See: Adachi and Brinkschulte, A global estimate for the Diederich–Fornæss index of weakly pseudoconvex domains, Preprint, 2014.

  2. We refer the reader to related work of Biard [2] which we became aware of after this work was completed.

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Acknowledgments

This work was done while the first author visited the University of Notre Dame in April, 2012. He thanks the Department of Mathematics for the warm hospitality. The authors also thank the referee for constructive suggestions. The authors were supported in part by NSF Grants.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Rutgers University-Camden, Camden, NJ, 08102, USA

    Siqi Fu

  2. Department of Mathematics, University of Notre Dame, Notre Dame, IN, 46556, USA

    Mei-Chi Shaw

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  1. Siqi Fu
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  2. Mei-Chi Shaw
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Correspondence to Siqi Fu.

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Fu, S., Shaw, MC. The Diederich–Fornæss Exponent and Non-existence of Stein Domains with Levi-Flat Boundaries. J Geom Anal 26, 220–230 (2016). https://doi.org/10.1007/s12220-014-9546-6

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  • DOI: https://doi.org/10.1007/s12220-014-9546-6

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