Tuschmann, Wilderich; Wiemeler, Michael Smooth stability and sphere theorems for manifolds and Einstein manifolds with positive scalar curvature. (English) Zbl 1426.53062 Commun. Anal. Geom. 27, No. 2, 491-509 (2019). The authors study smooth closed manifolds whose mean value of the scalar curvature is at least that of the unit sphere of the same dimension.Motivated by a rigidity result from [L. W. Green, Ann. Math. (2) 78, 289–299 (1963; Zbl 0116.13503)] for such manifolds whose conjuguate radius is equal to \(\pi\), they prove some “differential stability and sphere theorem versions” thereof.They first prove stability for such a manifold \(M\) of dimension \(n\) in the following sense: If the conjuguate radius \(\operatorname{Conj} M\) is sufficiently close to \(\pi\), then (under some additional technical assumptions), \(M\) is diffeomorphic to an \(n\)-dimensional spherical space form. To get that \(M\) is diffeomorphic to the standard unit \(n\)-sphere \(\mathbb{S}^n\), they further assume that the injectivity radius is close enough to \(\pi\).Then, as an application, the authors turn to closed \(n\)-dimensional manifolds of Ricci curvature at least \(n-1\). Under some limitation on the topology of \(M\) and technical assumptions, they prove that, if \(\operatorname{Conj} M\) is sufficiently close to \(\pi\), then \(M\) is diffeomorphic to an Einstein manifold with Einstein constant \(n-1\).They conjecture that such \(M\) should even be diffeomorphic to \(\mathbb{S}^{n}\), but prove it only when the sectional curvature is above a critical value.Their proofs use “\(C^{k,\alpha}\) convergence techniques developed in particular by [M. T. Anderson, Invent. Math. 102, No. 2, 429–445 (1990; Zbl 0711.53038)] and [M. T. Anderson and J. Cheeger, J. Differ. Geom. 35, No. 2, 265–281 (1992; Zbl 0774.53021)], and appropriate modifications of Green’s original arguments along with convergence properties of the injectivity and conjugate radius”. Reviewer: Julien Cortier (Montpellier) MSC: 53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 53C24 Rigidity results Keywords:closed manifold of positive scalar curvature; large conjuguate radius or injectivity radius; differential rigidity; differential stability Citations:Zbl 0116.13503; Zbl 0711.53038; Zbl 0774.53021 PDFBibTeX XMLCite \textit{W. Tuschmann} and \textit{M. Wiemeler}, Commun. Anal. Geom. 27, No. 2, 491--509 (2019; Zbl 1426.53062) Full Text: DOI arXiv