论文标题
$ n $ -sphere的船体的稳定性在$ \ mathbb {c}^n $中
Stability of the hull(s) of an $n$-sphere in $\mathbb{C}^n$
论文作者
论文摘要
我们研究了（全局）主教问题，用于$ \ mathbf {s}^n $ ---- $ \ Mathbb {C} \ times \ Mathbb {r}^{n-1} $ ----- in $ \ Mathbb {c}^n $。我们表明，如果$ s \ subset \ mathbb {c}^n $是$ \ mathbf {s}^n $（在$ \ mathcal {c}^3 $ -norm中）的足够小的扰动，那么$ s $ s $ bount a $（n+1）$ - n+1）。 $ s $的磁盘。此外，如果$ s $是平稳的或真正的分析性的，那么$ m $（直至其边界）。最后，如果$ s $是真正的分析（并且满足温和的状况），那么$ m $既是Holomorphy的信封，又是$ s $的多项式凸壳。这将先前已知的$ n = 2 $（cr奇点）概括为更高的维度（cr奇异性是非分离的）。
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).
