Let $\{ M_k \}_{k=1}^{\infty}$ be a sequence of closed, singular, minimal hypersurfaces in a closed Riemannian manifold $(N^{n+1},g), n+1 \geq 3$. Suppose, the volumes of $M_k$ are uniformly bounded from above and the $p$-th Jacobi eigenvalues of $M_k$ are uniformly bounded from below. Then, there exists a closed, singular, minimal hypersurface $M$ in $N$ with the above mentioned volume and eigenvalue bounds such that possibly after passing to a subsequence, $M_k$ weakly converges (in the sense of varifolds) to $M$, possibly with multiplicities. Moreover, the convergence is smooth and graphical over the compact subsets of $reg(M) \setminus Y$ where $Y$ is a finite subset of $reg(M)$ with $|Y|\leq p-1$. This result generalizes the previous results of Sharp and Ambrozio-Carlotto-Sharp in higher dimensions.

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Last updated: 06 Mar 2020