Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Valence of complex-valued planar harmonic functions

Author: Genevra Neumann
Journal: Trans. Amer. Math. Soc. 357 (2005), 3133-3167
MSC (2000): Primary 30C99, 26B99; Secondary 31A05, 26C99
Published electronically: December 2, 2004
MathSciNet review: 2135739
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Abstract: 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 \subseteq \mathbb{C} $. Let $S$ denote the critical set of $f$and $C(f)$ the global cluster set of $f$. We show that $f(S) \cup C(f)$ partitions the complex plane into regions of constant valence. We give some conditions such that $f(S) \cup C(f)$has empty interior. We also show that a component $R_0 \subseteq R \backslash f^{-1} (f(S) \cup C(f))$ is an $n_0$-fold covering of some component $\Omega_0 \subseteq \mathbb{C}\backslash (f(S) \cup C(f))$. If $\Omega_0$ is simply connected, then $f$ is univalent on $R_0$. We explore conditions for combining adjacent components to form a larger region of univalence. Those results which hold for $C^1$ functions on open sets in $\mathbb{R} ^2$ 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 $\mathbb{C}\backslash (f(S) \cup C(f))$sharing a common boundary arc in $f(S) \backslash C(f)$.

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Genevra Neumann
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
Address at time of publication: Department of Mathematics, Kansas State University, Manhattan, Kansas 66506

Keywords: Planar harmonic functions, $C^1$ functions in $\mathbb{R}^2$, regions of constant valence
Received by editor(s): September 17, 2003
Published electronically: December 2, 2004
Article copyright: © Copyright 2004 American Mathematical Society