Abstract
We study the (global) Bishop problem for small perturbations of Sn — the unit sphere of C×Rn−1 — in Cn. We show that if S⊂Cn is a sufficiently-small perturbation of Sn (in the C3-norm), then S bounds an (n+1)-dimensional ball M⊂Cn that is foliated by analytic disks attached to S. Furthermore, if S is either smooth or real analytic, then so is M (up to its boundary). Finally, if S is real analytic (and satisfies a mild condition), then M is both the envelope of holomorphy and the polynomially convex hull of S. This generalizes the previously known case of n=2 (CR singularities are isolated) to higher dimensions (CR singularities are nonisolated).
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