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 $\Omega =\mathbb {R}^{n}$).

关于 $C^{\infty }$ 度量，它接近 $\mathbb {R}^{N+1+\ell }$ 上的标准欧几里德度量，其中 $N\ge 7$ 和 $\ell \ge 1$ 给定，我们构造一类嵌入的 $(N+\ell )$ 维超曲面（无边界），它是最小且严格稳定的，并且奇异集等于任意预分配的封闭子集 $K\subset \{0\}\times \mathbb {R}^{\ell }$。因此，关于奇异集合中是否可以存在“间隙”或什至分数维部分的问题得到了强烈肯定。对于稳定和不稳定的最小子流形，此类问题在真实分析度量的情况下在所有维度上都保持开放，特别是对于标准欧几里得度量。