Stationary and stable varifolds with singularities

We construct minimal $m$-dimensional immersions in ${\mathbb{R}}^{m+1}$, equipped with a $C^{1,\alpha }$ metric, $\alpha \in [0,1)$, with a sequence of \emph{catenoidal necks} or \emph{floating disks} converging to an isolated, multiplicity $2$, singular flat point.