Generic scarring for minimal hypersurfaces along stable hypersurfaces

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