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

@article{Gupta2020StabilityOT, title={Stability of the hull(s) of an \$n\$-sphere in \$\mathbb\{C\}^n\$}, author={Purvi Gupta and Chloe Urbanski Wawrzyniak}, journal={arXiv: Complex Variables}, year={2020} }

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…

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