Compactness of certain class of singular minimal hypersurfaces

Abstract

Let $\{M_k\}_{k=1}^{\infty }$ be a sequence of closed, singular, minimal hypersurfaces in a closed Riemannian manifold $(N^{n+1},g), n+1 \ge 3$. Suppose, the volumes of $M_k$ are uniformly bounded from above and the pth Jacobi eigenvalues $\lambda _p$’s of $M_k$ are uniformly bounded from below. Then we will prove that 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 $reg(M) \setminus {\mathcal {Y}}$, where ${\mathcal {Y}}$ is a finite subset of reg(M) with $|{\mathcal {Y}}|\le p-1$. As a corollary, we get the compactness of the space of closed, singular, minimal hypersurfaces with uniformly bounded volume and index. These results generalize the previous theorems of Ambrozio–Carlotto–Sharp (J Geom Anal 26(4):2591–2601, 2016) and Sharp (J Differ Geom 106(2):317–339, 2017) in higher dimensions. We will also show that if $\varSigma$ is a singular, minimal hypersurface with ${\mathcal {H}}^{n-2}(sing(\varSigma ))=0$, then the index of the varifold associated to $\varSigma$ coincides with the index of $reg(\varSigma )$ (with respect to compactly supported normal vector fields on $reg(\varSigma )$).