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


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.


Bishop disc Complex tangency CR singularity Polynomially convex 

Mathematics Subject Classification (2000)

Primary 32E20 46J10 Secondary 30E10 


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  1. 1.
    Bharali G.: Surfaces with degenerate CR singularities that are locally polynomially convex. Michigan Math. J. 53, 429–445 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bharali, G.: Polynomial approximation, local polynomial convexity, and degenerate CR singularities II. Int. J. Math. (to appear), arxiv:1010.5205Google Scholar
  3. 3.
    Bishop E.: Differentiable manifolds in complex Euclidean space. Duke Math. J. 32, 1–21 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chirka E.M., Shcherbina N.V.: Pseudoconvexity of rigid domains and foliations of hulls of graphs. Ann. Scuola Norm. Sup. Pisa Cl. Sci. Ser. IV 22, 707–735 (1995)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Duval J., Sibony N.: Polynomial convexity, rational convexity, and currents. Duke Math. J. 79, 487–513 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Forstnerič F.: Analytic disks with boundaries in a maximal real submanifold of \({\mathbb{C}^2}\). Ann. Inst. Fourier 37, 1–44 (1987)CrossRefzbMATHGoogle Scholar
  7. 7.
    Forstnerič F., Stout E.L.: A new class of polynomially convex sets. Arkiv. Math. 29, 51–62 (1991)CrossRefzbMATHGoogle Scholar
  8. 8.
    Jöricke B.: Local polynomial hulls of discs near isolated parabolic points. Indiana Univ. Math. J. 46, 789–826 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kenig C.E., Webster S.: The local hull of holomorphy of a surface in the space of two complex variables. Invent. Math. 67, 1–21 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Merker, J., Porten, E.: Holomorphic extension of CR functions, envelopes of holomorphy, and removable singularities. Int. Math. Res. Surv. 2006, Art. ID 28925Google Scholar
  11. 11.
    Shcherbina N.V.: On the polynomial hull of a graph. Indiana Univ. Math. J. 42, 477–503 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Stout E.L.: Polynomial Convexity. Progress in Mathematics, vol. 261. Birkhaüser, London (2007)Google Scholar
  13. 13.
    Stout E.L.: Polynomially convex neighborhoods of hyperbolic points. Abstr. AMS 7, 174 (1986)Google Scholar
  14. 14.
    Wiegerinck J.: Locally polynomially convex hulls at degenerated CR singularities of surfaces in \({\mathbb{C}^2}\). Indiana Univ. Math. J. 44, 897–915 (1995)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of ScienceBangaloreIndia

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