Compactness of the space of minimal hypersurfaces with bounded volume and p-th Jacobi eigenvalue

TL;DR

This paper establishes a compactness theorem for minimal hypersurfaces in low-dimensional closed Riemannian manifolds, linking volume bounds and spectral conditions on the stability operator, with improved convergence properties under higher eigenvalue bounds.

Contribution

It extends compactness results by replacing the first eigenvalue bound with bounds on the p-th Jacobi eigenvalue, leading to stronger convergence properties.

Findings

Proves compactness for minimal hypersurfaces with volume and eigenvalue bounds.

Shows strong convergence away from finitely many points under p-th eigenvalue bounds.

Provides new spectral conditions for the regularity of minimal hypersurfaces.

Abstract

Given a closed Riemannian manifold of dimenion less than eight, we prove a compactness result for the space of closed, embedded minimal hypersurfaces satisfying a volume bound and a uniform lower bound on the first eigenvalue of the stability operator. When the latter assumption is replaced by a uniform lower bound on the p-th Jacobi eigenvalue for p greater or equal than 2 one gains strong convergence to a smooth limit submanifold away from at most p-1 points.