Generic Scarring for Minimal Hypersurfaces in Manifolds Thick at Infinity with a Thin Foliation at Infinity

We show generic scarring phenomenon for minimal hypersurfaces in a class of complete non-compact manifolds. In particular, we prove that for any metric $g$ in a $C^{\mathrm{\infty }}$-generic subset of the family of complete metrics which are thick at infinity with a thin foliation at infinity on a fixed $M^{n+1}$ of dimension $3\le (n+1)\le 7$, to any connected, closed, embedded, $2$-sided, stable minimal hypersurface $S\subset (M,g)$, there exists a sequence of closed, embedded, minimal hypersurfaces $\{{\mathrm{\Sigma }}_k\}$ scarring along $S$, in the sense that the area of ${\mathrm{\Sigma }}_k$ diverges to infinity, and when properly renormalized, ${\mathrm{\Sigma }}_k$ converges to $S$ as varifolds.