2004/01/26 by Genevra Neumann, Neumann, Genevra
Mathematics · #Analytic and geometric function theory #Holomorphic and Operator Theory #Numerical methods in inverse problems #math.CV #math.GT
paper · pdf · doi:10.48550/arxiv.math/0401359
31 pages, 10 figures. Question for geometers: Please email the author if you know of results similar to Theorem 3.4 in this paper
arxiv created 2004/01/26 · arxiv updated 2009/12/01
The valence of a function f at a point w is the number of distinct, finite solutions to f(z) = w. Let f be a complex-valued harmonic function in an open set R ⊆ ℂ. Let S denote the critical set of f and C(f) the global cluster set of f. We show that f(S) ∪ C(f) partitions the complex plane into regions of constant valence. We give some conditions such that f(S) ∪ C(f) has empty interior. We also show that a component R0 ⊆ R \backslash f-1(f(S) ∪ C(f)) is a n0-fold covering of some component Ω0 ⊆ ℂ \backslash (f(S) ∪ C(f)). If Ω0 is simply connected, then f is univalent on R0. We explore conditions for combining adjacent components to form a larger region of univalence. Those results which hold for C1 functions on open sets in ℝ2 are first stated in that form and then applied to the case of planar harmonic functions. If f is a light, harmonic function in the complex plane, we apply a structure theorem of Lyzzaik to gain information about the difference in valence between components of ℂ \backslash (f(S) ∪ C(f)) sharing a common boundary arc in f(S) \backslash C(f).