Mean curvature flows with prescribed singular sets

For every closed set $K\subset {\mathbb{R}}^n$ and every $m\ge 2$, we construct a mean-convex ancient solution to mean curvature flow of hypersurfaces in ${\mathbb{R}}^{m+n}$, with respect to a smooth Riemannian metric arbitrarily $C^{\mathrm{\infty }}$-close to the Euclidean metric, whose first-time singular set is exactly $K\times \{0\}$.