Bounds for the first Dirichlet eigenvalue of domains in Kaehler manifolds
type: Item
instance of: scholarly article
title: Bounds for the first Dirichlet eigenvalue of domains in Kaehler manifolds
zbMATH Open document ID: 0735.58009
DOI: 10.1007/BF01198223
author: Fernando Giménez
author: Vicente Miquel
published in: Archiv der Mathematik
publication date: 1992-06-25T00:00:00Z
review text: The authors study the first eigenvalue \(\mu_ 1(M)\) of the Dirichlet problem \[ \Delta f=\mu f\text{ on }M\text{ and }f=0\text{ on }\partial M, \] where \(M\) is a connected compact Kähler manifold of real dimension \(2n\) with smooth boundary \(\partial M\). They find some geometric conditions for \(M\) and \(\partial M\) to get a lower bound \(\mu_ 1(B^ \mathbb{C} _ r)\) for \(\mu_ 1(M)\). Here \(B^ \mathbb{C} _ r\) (resp. \(S^ \mathbb{C} _ r\)) denotes the geodesic ball (resp. sphere) of radius \(r\) in \(\mathbb{C} P^ n(\lambda)\). If \(\partial M\) is connected they prove that the equality \(\mu_ 1(M)=\mu_ 1(B^ \mathbb{C} _ r)\) is equivalent to some equality for \(r\). Under the same conditions it is proved there is a holomorphic isometry \(i: M\rightarrow B^ \mathbb{C} _ r\), such that \(i(\partial M)=S^ \mathbb{C} _ r=\partial B^ \mathbb{C} _ r\).
Mathematics Subject Classification ID: 58C40
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 12474
zbMATH Keywords: normal curvature
zbMATH Keywords: mean curvature
zbMATH Keywords: first eigenvalue
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: Kähler manifold
zbMATH Keywords: holomorphic isometry
reviewed by: Neda Bokan
MaRDI profile type: Publication
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cites work: Q3344779
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full work available at URL: https://doi.org/10.1007/bf01198223
OpenAlex ID: W2058424370
description: scientific article; zbMATH DE number 12474