On Gromov's compactness question regarding positive scalar curvature (Q6623109)

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.