Robust index bounds for minimal hypersurfaces of isoparametric submanifolds and symmetric spaces

We find many examples of compact Riemannian manifolds (M, g) whose closed minimal hypersurfaces satisfy a lower bound on their index that is linear in their first Betti number. Moreover, we show that these bounds remain valid when the metric g is replaced with g′ in a neighbourhood of g. Our examples (M, g) consist of certain minimal isoparametric hypersurfaces of spheres, their focal manifolds, the Lie groups SU (n) for n≤ 17 and Sp (n) for all n, and all quaternionic Grassmannians.