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Smooth stability and sphere theorems for manifolds and Einstein manifolds with positive scalar curvature. (English) Zbl 1426.53062

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”.

MSC:

53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53C24 Rigidity results
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