Generic scarring for minimal hypersurfaces along stable hypersurfaces

Abstract

Let $M^{n+1}$ be a closed manifold of dimension $3\le n+1\le 7$. We show that for a $C^{\infty }$-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 $\{{\Sigma }_k\}$ scarring along $S$, in the sense that the area and Morse index of ${\Sigma }_k$ both diverge to infinity and, when properly renormalized, ${\Sigma }_k$ converges to $S$ as varifolds. We also show that scarring of immersed minimal surfaces along stable surfaces occurs in most closed Riemannian $3$-manifods.

arXiv commentv2: final version, to appear in GAFA

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