Geometric Interpretation of the two dimensional Poisson Kernel and its applications

TL;DR

This paper revisits Hermann Schwarz's geometric interpretation of the 2D Poisson kernel, using it to develop new methods and results in hyperbolic geometry, including a novel One Radius Theorem for hyperbolic Laplacian eigenfunctions.

Contribution

It introduces a new geometric approach to classical problems in hyperbolic geometry, leading to the proof of the One Radius Theorem and related results.

Findings

Established a One Radius Theorem for hyperbolic Laplacian eigenfunctions.

Developed new geometric methods for classical hyperbolic problems.

Connected the Poisson kernel's geometry to eigenfunction behavior.

Abstract

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.