Generic scarring for minimal hypersurfaces along stable hypersurfaces

Abstract

Let Mⁿ⁺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 {Σₖ} scarring along S, in the sense that the area and Morse index of Σₖ both diverge to infinity and, when properly renormalized, Σₖ converges to S as varifolds. We also show that scarring of immersed minimal surfaces along stable surfaces occurs in most closed Riemannian 3-manifods.