First stability eigenvalue of singular minimal hypersurfaces in spheres

TL;DR

This paper extends eigenvalue estimates for minimal hypersurfaces in spheres to include singular cases, establishing bounds on the first stability eigenvalue and characterizing equality cases.

Contribution

It introduces a stability eigenvalue estimate for singular minimal hypersurfaces in spheres under the lpha-structural hypothesis, generalizing classical results.

Findings

Singular minimal hypersurfaces have first stability eigenvalue at most -2n.

Equality holds only for products of two round spheres.

The result generalizes Simons' estimate to the singular setting.

Abstract

In this short note we extend an estimate due to J. Simons on the first stability eigenvalue of minimal hypersurfaces in spheres to the singular setting. Specifically, we show that any singular minimal hypersurface in $S^{n+1}$, which is not totally geodesic and satisfies the \alpha-structural hypothesis, has first stability eigenvalue at most -2n, with equality if and only if it is a product of two round spheres. The equality case was settled independently in the classical setting by Wu and Perdomo.