带边界的simply connected紧致光滑流形上Beltrami流的典型场线和边界场线行为
Typical field lines of Beltrami flows and boundary field line behaviour of Beltrami flows on simply connected, compact, smooth manifolds with boundary
摘要：
研究了紧致连通流形上第一deRham上同调群为零的Beltrami流的边界场线性质。也就是说，我们表明，除了一个最无处密集的边界子集，贝尔特拉米场可能会消失，所有其他领域的边界线顺利嵌入$1$ -流形同构于$\mathbb{R}$，接近零点集的时间去$\pm \infty$。然后，我们下降的假设的紧凑性和消失的德拉姆上同调和证明，几乎每一个点上的给定的流形，通过该点的场线是一个非常数，周期轨道或非周期轨道来任意接近的起点，随着时间的推移到$\pm \infty$。在证明的过程中，我们将特别证明，在流形内部贝尔特拉米场消失的点的集合在费德勒意义下是可数$1$ -可求积的，因此特别具有至多为$1$的豪斯多夫维数。作为结果，我们得出结论，对于每一个本征场的旋度算子，对应于一个非零的本征值，总是存在一个节点域。
英文摘要：
We characterise the boundary field line behaviour of Beltrami flows on compact, connected manifolds with vanishing first de Rham cohomology group. Namely we show that except for an at most nowhere dense subset of the boundary, on which the Beltrami field may vanish, all other field lines at the boundary are smoothly embedded $1$-manifolds diffeomorphic to $\mathbb{R}$, which approach the zero set as time goes to $\pm \infty$. We then drop the assumptions of compactness and vanishing de Rham cohomology and prove that for almost every point on the given manifold, the field line passing through the point is either a non-constant, periodic orbit or a non-periodic orbit which comes arbitrarily close to the starting point as time goes to $\pm \infty$. During the course of the proof we will in particular show that the set of points at which a Beltrami field vanishes in the interior of the manifold is countably $1$-rectifiable in the sense of Federer and hence in particular has a Hausdorff dimension of at most $1$. As a consequence we conclude that for every eigenfield of the curl operator, corresponding to a non-zero eigenvalue, there always exists exactly one nodal domain.
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