Local polynomial hulls of discs near isolated parabolic points

Abstract

Let ∆ be a C 2 disc imbedded into C 2 with isolated parabolic point. The problem is considered whether sufficiently small closed neighbourhoods of this point on the disc are polynomially convex. This problem remained open after a classical paper of E. Bishop. We show that generically the index of the parabolic point is zero and the answer is yes. However, we show by an explicit example that in the index zero case the answer may be no, in contrast to what one would like to expect. In such a case for any small enough closed neighbourhood K on ∆ of the parabolic point K tr def = K ∩ ( K \ K) has the structure of an "onion". The "coats" of the onion bound analytic discs. Here K denotes the polynomial hull of K. Parabolic points of index +1 and -1 are also considered.

Local Polynomial Convexity of the Unfolded Whitney Umbrella in ℂ2

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Polynomially convex hulls of singular real manifolds

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