Recent developments in differentiable sphere theorems (Q2900300)

The paper represents a survey on the most important differentiable sphere theorems. It introduces recent developments in this field. Using the Ricci flow and stable currents, the author constructs an intrinsic invariant \(I(M)\) for an oriented complete Riemannian \(n\)-manifold \(M\) via a scalar quantity involving the scalar curvature and the mean curvature. The author proves that if \(I(M)>0\), then \(M\) is diffeomorphic to \(S^n\), for arbitrary \(n\). The paper contains a generalization of the Brendle-Schoen differentiable sphere theorem for manifolds with strictly \(1\over 4\)-pinched curvatures in the pointwise sense to the cases of submanifolds in a Riemannian manifold with codimension \(\geq0\). The author also proves several new differentiable sphere theorems for various pinched manifolds and presents a partial solution to Yau's open problem on pinching theorem.NEWLINENEWLINEFor the entire collection see [Zbl 1235.00045].