Topology of level curves of harmonic functions
Wilfred Kaplan · Transactions of the American Mathematical Society · 1948
By a regular curve-family(') is meant a family which is locally homeomorphic to a family of parallel lines. It follows from the theory of conformal mapping that D can always be chosen to be one of the two domains D1: the interior of the unit circle x2+y2 = 1, D.: the entire xy-plane. If D can be chosen as D1, F will be termed hyperbolic; if D can be chosen as D,,, F will be termed parabolic. The cases are not mutually exclusive, as the following theorem shows.