The Topology of the Level Curves of Harmonic Functions with Critical Points

Abstract

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