The local polynomial hull near a degenerate CR singularity -- Bishop discs revisited
Abstract
Let S be a smooth real surface in C^2 and let p\in S be a point at which the tangent plane is a complex line. How does one determine whether or not S is locally polynomially convex at such a p --- i.e. at a CR singularity ? Even when the order of contact of T_p(S) with 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(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
Cite
@article{arxiv.0902.4215,
title = {The local polynomial hull near a degenerate CR singularity -- Bishop discs revisited},
author = {Gautam Bharali},
journal= {arXiv preprint arXiv:0902.4215},
year = {2015}
}
Comments
20 pages; all theorems restated to admit surfaces of lower regularity; minor errors in Step 4 of Thm. 1.5 corrected; to appear in Math. Zeitschrift