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Abstract We find examples of cohomogeneity one metrics on$$S^4$$ and$$\mathbb {C}P^2$$ with positive sectional curvature that lose this property when evolved via Ricci flow. These metrics are arbitrarily small perturbations of Grove–Ziller metrics with flat planes that become instantly negatively curved under Ricci flow.
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Abstract We study relations between certain totally geodesic foliations of a closed flat manifold and its collapsed Gromov–Hausdorff limits. Our main results explicitly identify such collapsed limits as flat orbifolds, and provide algebraic and geometric criteria to determine whether they are singular.more » « less
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Following Gromov, a Riemannian manifold is called area-extremal if any modification that increases scalar curvature must decrease the area of some tangent 2-plane. We prove that large classes of compact 4-manifolds, with or without boundary, with nonnegative sectional curvature are area-extremal. We also show that all regions of positive sectional curvature on 4-manifolds are locally area-extremal. These results are obtained analyzing sections in the kernel of a twisted Dirac operator constructed from pairs of metrics, and using the Finsler–Thorpe trick for sectional curvature bounds in dimension 4.more » « less
