Compactness of certain class of singular minimal hypersurfaces

• Published in: Calculus of variations and partial differential equations 2022-02, Vol.61 (1), Article 24
• Main Author: Dey, Akashdeep
• Format: Article
• Language: English
• Subjects: Analysis; Calculus of Variations and Optimal Control; Optimization; Control; Convergence; Eigenvalues; Fields (mathematics); Hyperspaces; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Riemann manifold; Systems Theory; Theoretical
• ISSN: 0944-2669; 1432-0835
• DOI: 10.1007/s00526-021-02136-w

Let { M k } k = 1 ∞ be a sequence of closed, singular, minimal hypersurfaces in a closed Riemannian manifold ( N n + 1 , g ) , n + 1 ≥ 3 . Suppose, the volumes of M k are uniformly bounded from above and the p th Jacobi eigenvalues λ 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 r e g ( M ) \ Y , where Y is a finite subset of reg ( M ) with | Y | ≤ 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 Σ is a singular, minimal hypersurface with H n - 2 ( s i n g ( Σ ) ) = 0 , then the index of the varifold associated to Σ coincides with the index of r e g ( Σ ) (with respect to compactly supported normal vector fields on r e g ( Σ ) ).