Minimal hypersurfaces with cylindrical tangent cones

First we construct minimal hypersurfaces $M\subset\mathbf{R}^{n+1}$ in a neighborhood of the origin, with an isolated singularity but cylindrical tangent cone $C\times \mathbf{R}$, for any strictly minimizing strictly stable cone $C$ in $\mathbf{R}^n$. We show that many of these hypersurfaces are area minimizing. Next, we prove a strong unique continuation result for minimal hypersurfaces $V$ with such a cylindrical tangent cone, stating that if the blowups of $V$ centered at the origin approach $C\times \mathbf{R}$ at infinite order, then $V = C\times\mathbf{R}$ in a neighborhood of the origin. Using this we show that for quadratic cones $C = C(S^p \times S^q)$, in dimensions $n > 8$, all $O(p+1) \times O(q+1)$-invariant minimal hypersurfaces with tangent cone $C\times \mathbf{R}$ at the origin are graphs over one of the surfaces that we constructed. In particular such an invariant minimal hypersurface is either equal to $C\times \mathbf{R}$ or has an isolated singularity at the origin.