Harmonic forms on manifolds with weighted Poincaré inequality
type: Item
instance of: scholarly article
title: Harmonic forms on manifolds with weighted Poincaré inequality
zbMATH Open document ID: 1177.58006
author: Jui-Tang Ray Chen
author: Chiung-Jue Anna Sung
published in: Pacific Journal of Mathematics
publication date: 2009-10-02T00:00:00Z
review text: Let \(M^ m\) be an \(m\)-dimensional complete Riemannian manifold. \(M^ m\) is said to satisfy a weighted Poincaré inequality with a non-negative weight function \(\rho(x)\) if the inequality \(\int_ M\rho(x)\varphi^ 2\leq \int_ M|\nabla\varphi|^ 2\) is valid for all compactly supported smooth functions \(\varphi\). Let \(H^ p_ d(M^ m)\) denote the space of \(L^ d\) harmonic \(p\)-forms on \(M^ m\). In this paper, the authors study general harmonic \(p\)-forms. They assume both the curvature operator lower bound expressed in terms of the dimension and the weighted Poincaré inequality with the weight function \(\rho\) on \(M^ m\) to prove vanishing and finiteness theorems for the \(L^ d\) harmonic \(p\)-forms from \(H^ p_ d(M^ m)\).
reviewed by: Andrew Bucki
Mathematics Subject Classification ID: 58A10
Mathematics Subject Classification ID: 53C21
zbMATH DE Number: 5610413
zbMATH Keywords: weighted Poincaré inequality
zbMATH Keywords: harmonic forms
MaRDI profile type: Publication
full work available at URL: pjm.2009.242.201
OpenAlex ID: W2051314661
DOI: 10.2140/PJM.2009.242.201
description: scientific article; zbMATH DE number 5610413