Geometric Interpretation of the two dimensional Poisson Kernel and its applications

2009/12/01 by Sergei Artamoshin, Artamoshin, Sergei · 2 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #math.DG

paper · pdf · doi:10.48550/arxiv.0912.0223

147 pages, 23 PostScript figures, phd thesis

arxiv created 2009/12/01 · openalex publication_date 2009/12/01 · arxiv updated 2014/11/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• The first Dirichlet Eigenvalue of a Compact Manifold and the Yang Conjecture

Cited by

• The Spherical Ratio of Two Points and its Integral Properties
• Lower Bounds For The First Eigenvalue Of The Laplacian With Dirichlet Boundary Conditions In A Hyperbolic Space Of A Negative Constant Curvature

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