Extremal properties of the principal Dirichlet eigenvalue for regular polygons in the hyperbolic plane
type: Item
instance of: scholarly article
title: Extremal properties of the principal Dirichlet eigenvalue for regular polygons in the hyperbolic plane
zbMATH Open document ID: 1034.52013
author: Leon Karp
author: Norbert Peyerimhoff
published in: Archiv der Mathematik
publication date: 2002-12-01T00:00:00Z
review text: The authors show that, in the hyperbolic plane, the equilateral triangle is the unique triangle with minimal principal Dirichlet eigenvalue amongst the hyperbolic triangles of equal finite perimeter. The proof is based on a generalization of Steiner symmetrization to the hyperbolic plane and on the fact that the convex hull of a symmetrized hyperbolic triangle is still a triangle. Moreover, the paper describes the locus of all points \(P\) such that, for two given points \(P_1\) and \(P_2\) in the hyperbolic plane, the triangles with vertices, \(P_1, P_2\) and \(P\) have a prescribed area. Besides, the authors identify the unique hyperbolic quadrilateral with minimal principal Dirichlet eigenvalue amongst those in a given geodesic ball.
reviewed by: Maria do Rosário Pinto
Mathematics Subject Classification ID: 52B60
Mathematics Subject Classification ID: 51M10
zbMATH DE Number: 1837332
zbMATH Keywords: hyperbolic triangles
zbMATH Keywords: Dirichlet eigenvalue
zbMATH Keywords: Steiner symmetrization
MaRDI profile type: Publication
full work available at URL: s00013-002-8308-z
OpenAlex ID: W1963682167
DOI: 10.1007/S00013-002-8308-Z
description: scientific article; zbMATH DE number 1837332