Stable minimal hypersurfaces in ${\mathbb{R}}^{N+1+\ell }$ with singular set an arbitrary closed $K$ in $\{0\}\times {\mathbb{R}}^{\ell }$

Abstract

With respect to a $C^{\infty }$ metric which is close to the standard Euclidean metric on ${\mathbb{R}}^{N+1+\ell }$, where $N\ge 7$ and $\ell \ge 1$ are given, we construct a class of embedded $(N+\ell )$-dimensional hypersurfaces (without boundary) which are minimal and strictly stable, and which have singular set equal to an arbitrary preassigned closed subset $K\subset \{0\}\times {\mathbb{R}}^{\ell }$.

Expository improvements. To appear in Annals of Math