Stability and the first Betti number

Abstract

This short, expository note focuses on the index of compact, minimal hypersurfaces of a sphere. After reviewing the main facts, we announce, without proof, a comparison theorem between the spectrum of the stability operator of such immersions and that of the Laplacian on 1-forms. The geometric consequence is a lower bound of the index by the first Betti number of the hypersurface which implies that, if the first Betti number is large, then the immersion is highly unstable. Proofs will appear elsewhere.