Examples of simply-connected Liouville manifolds with positive spectrum
type: Item
instance of: scholarly article
title: Examples of simply-connected Liouville manifolds with positive spectrum
zbMATH Open document ID: 0861.53033
DOI: 10.1016/0926-2245(96)00005-8
author: Itai Benjamini
author: Jianguo Cao
published in: Differential Geometry and its Applications
publication date: 1996-08-05T00:00:00Z
review text: Let \(n\geq 3\). The authors present a family of Riemannian metrics \(g\) on Euclidean space \(\mathbb{R}^n\) so that \((\mathbb{R}^n,g)\) has positive bottom of the spectrum of the Laplacian, i.e., \(\lambda_1(\mathbb{R}^n,g)>0\), and bounded geometry \(|K|\leq C\), but \((\mathbb{R}^n,g)\) has no nonconstant bounded harmonic functions. This gives a negative answer to a problem posed by \textit{R. Schoen} and \textit{S.-T. Yau} [Lectures on differential geometry. Cambridge, MA: International Press (1994; Zbl 0830.53001)]. If \(n=3\), these metrics are double warped product metrics of the form \(g=e^{2\eta_1(s)}dx^2+e^{2\eta_2(s)}dy^2+ds^2\) where the \(\eta_i\) are suitably chosen.
reviewed by: Peter B. Gilkey
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 58J65
Mathematics Subject Classification ID: 60D05
zbMATH DE Number: 897464
zbMATH Keywords: positive spectrum
zbMATH Keywords: Liouville manifolds
zbMATH Keywords: bounded geometry
zbMATH Keywords: Brownian motion
zbMATH Keywords: bounded harmonic functions
Wikidata QID: Q115362525
MaRDI profile type: Publication
full work available at URL: 0926-2245(96)00005-8
OpenAlex ID: W2086223996
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description: scientific article; zbMATH DE number 897464