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 (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).

我们研究（全局）Bishop 问题的小扰动 ${\mathbf{秒}}^n$ — 的单位球面 $\mathbb{C}\times {\mathbb{电阻}}^{n-1}$ - 在 ${\mathbb{C}}^n$. 我们证明如果$秒\subset {\mathbb{C}}^n$ 是一个足够小的扰动 ${\mathbf{秒}}^n$ （在里面 ${\mathcal{C}}^3$-norm)，然后S限定一个$(n+1)$次元球 $米\subset {\mathbb{C}}^n$由附加到S 的解析圆盘组成。此外，如果S是平滑的或实解析的，那么M（直到其边界）也是如此。最后，如果S是实解析的（并且满足温和条件），则M既是全纯的包络又是S的多项式凸包。这概括了先前已知的情况$n=2$ （CR 奇点是孤立的）到更高维度（CR 奇点是非孤立的）。