Stability of the hull(s) of an $n$-sphere in ${\mathbb{C}}^n$

Abstract

We study the (global) Bishop problem for small perturbations of ${\mathbf{S}}^n$ --- the unit sphere of $\mathbb{C}\times {\mathbb{R}}^{n-1}$ --- in ${\mathbb{C}}^n$. We show that if $S\subset {\mathbb{C}}^n$ is a sufficiently-small perturbation of ${\mathbf{S}}^n$ (in the ${\mathcal{C}}^3$-norm), then $S$ bounds an $(n+1)$-dimensional ball $M\subset {\mathbb{C}}^n$ that is foliated by analytic disks attached to $S$. Furthermore, if $S$ is either smooth or real analytic, then so is $M$ (upto 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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