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arXiv · 2006.03038

Generic scarring for minimal hypersurfaces along stable hypersurfaces

Antoine Song·Xin Zhou
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Abstract

Let $M^{n+1}$ be a closed manifold of dimension $3\leq n+1\leq 7$. 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.

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Antoine Song, Xin Zhou. 2020-06-04. Generic scarring for minimal hypersurfaces along stable hypersurfaces. https://arxiv.org/abs/2006.03038

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