Rigidity of riemannian manifolds containing an equator
In this paper, we prove that a Riemannian -manifold with sectional curvature bounded above by that contains a minimal -sphere of area which has index at least has constant sectional curvature . The proof uses the construction of ancient mean curvature flows that flow out of a minimal submanifold. As a consequence we also prove a rigidity result for the Simon-Smith minimal spheres.