Generic scarring for minimal hypersurfaces along stable hypersurfaces

TL;DR

This paper demonstrates that in generic metrics on certain manifolds, sequences of minimal hypersurfaces can be constructed to concentrate along stable hypersurfaces, with diverging area and index, and converge to the stable hypersurface as varifolds.

Contribution

It introduces a generic scarring construction for minimal hypersurfaces along stable ones, extending to immersed surfaces in 3-manifolds.

Findings

Sequences of minimal hypersurfaces with diverging area and index can be made to converge to stable hypersurfaces.

Scarring phenomena occur generically in high-dimensional manifolds and in most 3-manifolds.

The results apply to generic metrics and include both embedded and immersed minimal surfaces.

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.