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

TL;DR

This paper establishes linear lower bounds on the index of minimal hypersurfaces in certain compact Riemannian manifolds, including isoparametric hypersurfaces, Lie groups, and Grassmannians, with bounds stable under metric perturbations.

Contribution

It introduces new index bounds for minimal hypersurfaces in specific symmetric spaces and demonstrates their stability under metric changes.

Findings

Index bounds are linear in the first Betti number.

Bounds apply to various symmetric spaces and Lie groups.

Bounds remain valid under small metric perturbations.

Abstract

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 $n\leq 17$, and $Sp(n)$ for all $n$; and all quaternionic Grassmannians.