斯泰因流形中的光滑列维平坦超曲面的边界问题

Hanlong Fang, Xiaojun Huang, Wanke Yin, Zhengyi Zhou
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引用次数: 0

摘要

本文关注的问题是在$\mathbb C^{n+1}$或更广义的斯坦流形中,构造一个局部或全局附着于实数维二子流形的光滑列维-弗拉基表面,该流形具有椭圆CR奇异性,这一研究方向源于毕夏普(E. Bishop)的一篇基本经典论文。许多著名数学家在复分析和几何方面的早期研究都是沿着这个方向进行的。我们证明了一个紧凑光滑(或实解析)实码元二子曼形$M$,包含在一个平滑有界强伪凸域的边界中,在CR点有一个自然的必要条件,称为CR非最小条件,并且有两个椭圆CR奇异点,与一个平滑上界(分别为实解析上界)Levi平超曲面$\widehat{M}$相包围。这回答了多尔博-托马西尼-扎伊采夫工作中留下的一个众所周知的问题,或者说是毕肖普在 1965 年提出的一个问题的一般化版本。我们在这里的研究揭示了若干复杂分析与其他领域(如交映几何和折射理论)之间错综复杂的相互作用。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Bounding smooth Levi-flat hypersurfaces in a Stein manifold
This paper is concerned with the problem of constructing a smooth Levi-flat hypersurface locally or globally attached to a real codimension two submanifold in $\mathbb C^{n+1}$, or more generally in a Stein manifold, with elliptic CR singularities, a research direction originated from a fundamental and classical paper of E. Bishop. Earlier works along these lines include those by many prominent mathematicians working both on complex analysis and geometry. We prove that a compact smooth (or, real analytic) real codimension two submanifold $M$, that is contained in the boundary of a smoothly bounded strongly pseudoconvex domain, with a natural and necessary condition called CR non-minimal condition at CR points and with two elliptic CR singular points bounds a smooth-up-to-boundary (real analytic-up-to-boundary, respectively) Levi-flat hypersurface $\widehat{M}$. This answers a well-known question left open from the work of Dolbeault-Tomassini-Zaitsev, or a generalized version of a problem already asked by Bishop in 1965. Our study here reveals an intricate interaction of several complex analysis with other fields such as symplectic geometry and foliation theory.
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