Volume growth and positive scalar curvature
type: Item
instance of: scholarly article
title: Volume growth and positive scalar curvature
zbMATH Open document ID: 0979.53033
DOI: 10.1007/PL00001639
author: Guihua Gong
author: Guoliang Yu
published in: Geometric and Functional Analysis. GAFA
publication date: 2002-02-17T00:00:00Z
review text: The authors verify a conjecture of M. Gromov, namely that a macroscopically large Riemannian manifold cannot have uniformly positive scalar curvature [see \textit{M. Gromov}, Prog. Math. 132, 1-213 (1996; Zbl 0945.53022)], for Riemannian spin manifolds \( M \) with subexponential volume growth. The proof is based on the computation of the higher index of the Dirac operator on \(M\) and on a result of \textit{J. Roe} [J. Differ. Geom. 27, 87-113 (1988; Zbl 0657.58041), Proposition 4.33]. In particular, they prove Gromov's conjecture that a uniformly contractible complete Riemannian manifold \( (M,g)\) (i.e. for every \( r > 0\), there exists \( R>r\) such that every ball with radius \( r \) can be contracted to a point in the ball with radius \(R\)) with bounded geometry (i.e. \((M,g)\) has bounded sectional curvature and positive injectivity radius) cannot have uniformly positive scalar curvature when \((M, g)\) has subexponential volume growth.
reviewed by: Mircea Craioveanu
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C27
zbMATH DE Number: 1545122
zbMATH Keywords: uniformly contractible Riemannian manifold
zbMATH Keywords: macroscopically large Riemannian manifold
zbMATH Keywords: Roe's coarse homology theory
zbMATH Keywords: bounded geometry
zbMATH Keywords: subexponential volume growth
zbMATH Keywords: Riemannian spin geometry
zbMATH Keywords: Dirac operator
zbMATH Keywords: positive scalar curvature
zbMATH Keywords: hyperspherical manifold
zbMATH Keywords: Baum-Connes conjecture
zbMATH Keywords: Gromov's conjecture
zbMATH Keywords: zero-in-the-spectrum conjecture
MaRDI profile type: Publication
full work available at URL: pl00001639
OpenAlex ID: W2008449445
description: scientific article; zbMATH DE number 1545122