Compactness of certain class of singular minimal 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.