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

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 $K\subset \{0\}\times \mathbb {R}^{\ell }$. Thus the question is settled, with a strong affirmative, as to whether there can be “gaps” or even fractional dimensional parts in the singular set. Such questions, for both stable and unstable minimal submanifolds, remain open in all dimensions in the case of real analytic metrics and, in particular, for the standard Euclidean metric. \par The construction used here involves the analysis of solutions $u$ of the symmetric minimal surface equation on domains $\Omega \subset \mathbb {R}^{n}$ whose symmetric graphs (i.e., $\{(x,\xi )\in \Omega \times \mathbb {R}^{m}: |\xi |=u(x)\}$) lie on one side of a cylindrical minimal cone including, in particular, a Liouville type theorem for complete solutions (i.e., the case $\Omega =\mathbb {R}^{n}$).