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 \(reg(\varSigma )\)).