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Mathematische Zeitschrift

, Volume 271, Issue 3–4, pp 1043–1063 | Cite as

The local polynomial hull near a degenerate CR singularity: Bishop discs revisited

  • Gautam BharaliEmail author
Article

Abstract

Let \({\mathcal{S}}\) be a smooth real surface in \({\mathbb{C}^2}\) and let \({p\in\mathcal{S}}\) be a point at which the tangent plane is a complex line. How does one determine whether or not \({\mathcal{S}}\) is locally polynomially convex at such a p—i.e. at a CR singularity? Even when the order of contact of \({T_p(\mathcal{S})}\) with \({\mathcal{S}}\) at p equals 2, no clean characterisation exists; difficulties are posed by parabolic points. Hence, we study non-parabolic CR singularities. We show that the presence or absence of Bishop discs around certain non-parabolic CR singularities is completely determined by a Maslov-type index. This result subsumes all known facts about Bishop discs around order-two, non-parabolic CR singularities. Sufficient conditions for Bishop discs have earlier been investigated at CR singularities having high order of contact with \({T_p(\mathcal{S})}\). These results relied upon a subharmonicity condition, which fails in many simple cases. Hence, we look beyond potential theory and refine certain ideas going back to Bishop.

Keywords

Bishop disc Complex tangency CR singularity Polynomially convex 

Mathematics Subject Classification (2000)

Primary 32E20 46J10 Secondary 30E10 

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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of ScienceBangaloreIndia

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