论文标题
实际分析Beltrami字段的零集结构
Zero set structure of real analytic Beltrami fields
论文作者
论文摘要
在本文中，我们证明了真实分析性Beltrami字段零集的分类定理。即，我们表明，在删除其孤立点后，没有边界的真实分析，连接的$ 3 $ - manifold在没有边界的$ 3 $ - manifold上的零集是空的，或者可以将其写成可计数的，可算是有限的，具有分化的，连接的$ 1 $ $ -Dimensional submanifolds的（可能是空无一人）的边界和tame knoubnouss和tame knewots nounce knebounde。此外，我们考虑了这些驯服结的复杂性问题。 To this end we prove that on the standard (open) solid toroidal annulus in $\mathbb{R}^3$, there exist for any pair $(p,q)$ of positive, coprime integers countable infinitely many distinct real analytic metrics such that for each such metric there exists a real analytic Beltrami field, corresponding to the eigenvalue $+1$ of the curl operator, whose zero set is精确地由标准$（p，q）$ - 圆环结给出。指标和相应的Beltrami字段是明确构造的，可以单独通过基本功能在笛卡尔坐标中写下。
In this paper we prove a classification theorem for the zero sets of real analytic Beltrami fields. Namely, we show that the zero set of a real analytic Beltrami field on a real analytic, connected $3$-manifold without boundary is either empty after removing its isolated points or can be written as a countable, locally finite union of differentiably embedded, connected $1$-dimensional submanifolds with (possibly empty) boundary and tame knots. Further we consider the question of how complicated these tame knots can possibly be. To this end we prove that on the standard (open) solid toroidal annulus in $\mathbb{R}^3$, there exist for any pair $(p,q)$ of positive, coprime integers countable infinitely many distinct real analytic metrics such that for each such metric there exists a real analytic Beltrami field, corresponding to the eigenvalue $+1$ of the curl operator, whose zero set is precisely given by a standard $(p,q)$-torus knot. The metrics and the corresponding Beltrami fields are constructed explicitly and can be written down in Cartesian coordinates by means of elementary functions alone.
