Generic scarring for minimal hypersurfaces along stable hypersurfaces

• Published in: Geometric and functional analysis Vol. 31; no. 4; pp. 948 - 980
• Main Authors: Song, Antoine, Zhou, Xin
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.08.2021; Springer Nature B.V
• Subjects: Analysis; Hyperspaces; Mathematics; Mathematics and Statistics; Minimal surfaces; Scars
• ISSN: 1016-443X; 1420-8970
• DOI: 10.1007/s00039-021-00571-7

Let M n + 1 be a closed manifold of dimension 3 ≤ n + 1 ≤ 7 . We show that for a C ∞ -generic metric g on M , to any connected, closed, embedded, 2-sided, stable, minimal hypersurface S ⊂ ( M , g ) corresponds a sequence of closed, embedded, minimal hypersurfaces { Σ k } scarring along S , in the sense that the area and Morse index of Σ k both diverge to infinity and, when properly renormalized, Σ k converges to S as varifolds. We also show that scarring of immersed minimal surfaces along stable surfaces occurs in most closed Riemannian 3-manifods.