| Let K be a nonseparating continuum in the complex plane C and let f:C\D →C\K be the Riemann map from the complement of the closed unit disk onto the complement of K. The Green's function G : C → [0, infinity) is defined as Gz=&cubl4; logf-1 zif z∈C\K 0ifz∈K .; In this dissertation we develop a theory of Riemann maps and Green's functions which are dependent on a distance function rather than the conformal shape of K. The study of such a theory naturally leads to the study of the equidistant set of two sets X and Y. For two disjoint nonempty closed sets X and Y in C , the equidistant set of X and Y, denoted by E(X, Y), is defined by EX,Y=z∈ C:dz,X =dz,Y .; In the case that X and Y are two noninterlaced closed sets, we will show that their equidistant set is a one-manifold. We will also generalize the notion of an equidistant set for nonclosed sets. |
