The Topology of the Level Curves of Harmonic Functions with Critical Points
The Topology of the Level Curves of Harmonic Functions with Critical Points
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具有临界点的调和函数的电平曲线的拓扑
DOI：
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发表时间：
1951
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影响因子：
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通讯作者：
W. Boothby
中科院分区：
文献类型：
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作者：
W. Boothby
导言。在以前的一篇文章中，本文研究了填充欧几里得平面7r或r中的单连通区域的曲线族的拓扑性质。这些族被假定为正则的(即局部同胚于平行线)，除非在可能无限的孤立奇点集合上，每个族具有多鞍点的结构；这样的族被称为分枝正则曲线族。进一步研究这些族，特别是它们与调和函数的关系，是本文的目的。在下文中，将假定文献[I]中的定义和定理，并将使用相同的符号。特别地，F，G将表示填充平面7r的分枝正则曲线族，B将表示奇点集，R表示其中F为正则的域7rB，依此类推。欧几里得平面将被作为所有单连通区域的模型。文[1]的主要结果是证明了任何充满Xr的分支正则曲线族F都可以作为函数f(P)的水平曲线族给出，且f(P)在所有7r上连续且没有相对极值。这推广了[II]的一部分，其中证明了7r中没有奇点的曲线族也有相同的定理。本文的主要结果有两个：第一节证明了F实际上同胚于调和函数的水平曲线；第二节证明了F分解为一个可数曲线族集合，每个曲线族的结构为上半平面的平行线y=常数。这样的子族将被称为半平行，这种分解对下面将提到的调和函数和解析函数的研究有影响。这两个结果推广了
Introduction. In a previous paper,2 of which this is a continuation, topological properties of curve families which filled the Euclidean plane 7r, or a simply connected domain in r, were investigated. The families were assumed regular (i. e. locally homeomorphic to parallel lines) except at a possibly infinite collection of isolated singularities at each of which the family had the structure of a multiple saddle point; such families were called branched regular curve families. Further investigation of these families, in particular their relation to harmonic functions, is the aim of this paper. In what follows the definitions and theorems in [I] will be assumed, and the same notation will be used. In particular F, G will denote branched regular curve families filling the plane 7r, B will denote the set of singular points, R the domain 7r B in which F is regular, and so on. The Euclidean plane will be taken as a model for all simply connected domains. The principal result of [I] was to prove that any branched regular curve family F filling Xr can be given as the family of level curves of a function f (p) which is continuous on all of 7r and has no relative extrema. This generalizes a portion of [II] in which the same theorem is proved for a curve family without singularities in 7r. In this paper there are two main results: the first, proved in Section 1, is that F is actually homeomorphic to the level curves of a harmonic function; the second, proved in Section 2, asserts the existence of a decomposition of F into a countable collection of subfamilies of curves, each of which has the structure of the parallel lines y=constant of the upper half-plane. Such subfamilies will be called half-parallel, and this decomposition has consequences for the study of harmonic functions and analytic functions which will be mentioned below. These two results generalize