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 $\mathbb{C}\times\mathbb{R}^{n1}$  in $\mathbb{C}^n$. We show that if $S\subset\mathbb{C}^n$ is a sufficientlysmall 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).
 Publication:

arXiv eprints
 Pub Date:
 February 2020
 arXiv:
 arXiv:2002.08699
 Bibcode:
 2020arXiv200208699G
 Keywords:

 Mathematics  Complex Variables;
 32D10;
 32E20;
 32V25;
 32V40
 EPrint:
 A reference has been added