Geometric Interpretation of the two dimensional Poisson Kernel and its applications

Hermann Schwarz, while studying complex analysis, introduced the geometric interpretation for the Poisson kernel in 1890. We shall see here that the geometric interpretation can be useful to develop a new approach to some old classical problems as well as to obtain several new results, mostly related to hyperbolic geometry. For example, we obtain One Radius Theorem saying that any two radial eigenfunctions of a Hyperbolic Laplacian assuming the value 1 at the origin can not assume any other common value within some interval [0, p], where the length of this interval depends only on the location of the eigenvalues on the complex plane and does not depend on the distance between them.