Compactness of certain class of singular minimal hypersurfaces

TL;DR

This paper proves the compactness of a class of singular minimal hypersurfaces in a closed Riemannian manifold under uniform bounds on volume and the p-th Jacobi eigenvalue, extending previous results to higher dimensions.

Contribution

It generalizes existing compactness results for minimal hypersurfaces to include singular cases in higher-dimensional manifolds.

Findings

Compactness of the space of singular minimal hypersurfaces with bounded volume.

Lower bounds on the p-th Jacobi eigenvalue ensure compactness.

Extension of prior results to higher dimensions and singular hypersurfaces.

Abstract

Given a closed Riemannian manifold $(N^{n+1},g)$, $n+1 \geq 3$ we prove the compactness of the space of singular, minimal hypersurfaces in $N$ whose volumes are uniformly bounded from above and the $p$-th Jacobi eigenvalue $\lambda_p$'s are uniformly bounded from below. This generalizes the results of Sharp and Ambrozio-Carlotto-Sharp in higher dimensions.