On Gromov's compactness question regarding positive scalar curvature
type: Item
instance of: scholarly article
title: On Gromov's compactness question regarding positive scalar curvature
author: Shmuel Weinberger
author: Zhizhang Xie
author: Guoliang Yu
published in: Annals of \(K\)-Theory
publication date: 2024-10-23T00:00:00Z
review text: In this paper, the authors give both negative and positive examples to Gromov's compactness question [\textit{M Gromov}, Perspectives in scalar curvature, vol. 1. World Scientific. 1--514 (2023; Zbl 1532.53003)]: Let \(X\) be a smooth manifold. Suppose for any given compact subset \(V \subset X\) and any \(\rho > 0\), there exists a (noncomplete) Riemannian metric on \(X\) with scalar curvature \(\geq 1\) such that the closed \(\rho\)-neighborhood \(N_{\rho}(V)\) of \(V\) in \(X\) is compact. Then \(X\) admits a complete Riemannian metric with scalar curvature \(\geq 1\).\N\NIn order to give negative examples, they improve upon the methods to \textit{S. Chang} et al. [J. Geom. Phys. 149, Article ID 103575, 22 p. (2020; Zbl 1437.58017)] (see Section 3). The answer naturally leads us to a modification of Gromov's compactness question that a certain \(\varprojlim^{1}\) index-theoretic invariants vanishes (see Conjecture 4.1). They prove (Theorem 4.4) that the conjecture for \(1\)-tame spin manifolds of dimension \(\geq 6\) in which the pair \((\pi_{1}(M) , \pi_{1}^{\infty}(M))\) holds the unstable relative Gromov-Lawson-Rosenberg conjecture, where \(\pi_{1}^{\infty}(M)\) is the fundamental groupoid at infinity of \(M\) (see Definition 4.2). This result is also positive examples of Gromov's compactness question.
reviewed by: Tatsuki Seto
Mathematics Subject Classification ID: 58B34
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53C23
zbMATH DE Number: 7930661
zbMATH Keywords: Gromov's compactness conjecture on scalar curvature
zbMATH Keywords: index invariants at infinity
MaRDI profile type: Publication
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DOI: 10.2140/AKT.2024.9.583
description: scientific article; zbMATH DE number 7930661