Index estimate by first Betti number of minimal hypersurfaces in compact symmetric spaces

TL;DR

This paper establishes a lower bound on the Morse index of unstable minimal hypersurfaces in compact symmetric spaces, linking it to the first Betti number, and introduces a new approach for index estimation.

Contribution

It provides a novel method to estimate the Morse index of minimal hypersurfaces in symmetric spaces based on topological invariants.

Findings

Morse index is bounded below by a constant times the first Betti number

New approach extends previous methods for index estimation

Applicable to unstable minimal hypersurfaces in compact symmetric spaces

Abstract

We show that the Morse index of unstable closed minimal hypersurface $\Sigma$ in a compact semi-simple Riemannian symmetric space $M=G/K$ is bounded from below by constant times the first Betti number of $\Sigma$. Our proof is based on a natural extension of the previous method and this also provides a novel approach for the index estimate.