Index of Dirac operator and scalar curvature almost non-negative manifolds (Q2565978)

Let \(M\) be a closed spin manifold. It is known that \(M\) admits a metric with positive scalar curvature only if the index of the Dirac operator is zero. By [\textit{S. Stolz}, Ann. Math. (2) 136, No.3, 511--540 (1992; Zbl 0784.53029)], if \(M\) is simply connected and of dimension bigger than \(4\), \(M\) admits a metric of positive scalar curvature if and only if the index of the Dirac operator vanishes. \(M\) is said to be of almost non-negative scalar curvature if for any constant \(\epsilon > 0\), there is a Riemannian metric \(g\) on \(M\) such that \(s_g \, \text{ diam} (M, g)^2 \geq - \epsilon\) and \(\text{Sec}_g \leq 1\), where \(\text{Sec}_g\), \(s_g\) and \(\text{ diam} (M, g)\) denote the sectional curvature, the scalar curvature of \(g\) and the diameter of \((M, g)\), respectively. In the present paper, the author obtains a characterization of the manifolds with almost non-negative scalar curvature, proving that there are nontrivial topological obstructions for the existence of such manifolds. If the index of the Dirac operator does not vanish, the author proves that there exists a constant \(\varepsilon(n)>0\) such that if the scalar curvature \(s_M \geq -\varepsilon(n)\), then the fundamental group of \(M\) is finite. Moreover, \(M\) admits a real analytic Ricci-flat metric such that its Riemannian universal covering is isometric to the product of Ricci-flat Kähler-Einstein manifolds and/or Joyce manifolds with special holonomy group \(\text{ Spin}(7)\).