For each large enough $m\in\mathbb{N}$ we construct by PDE gluing methods a closed embedded smooth minimal hypersurface ${\breve{M}_m}$ doubling the equatorial three-sphere $\mathbb{S_{eq}}^3$ in $\mathbb{S}^4(1)$. This answers a long-standing question of Yau in the case of $\Sph4(1)$ and long-standing questions of Hsiang. Similarly for each integer or half-integer $J$, we construct a self-shrinker $\breve{M}[m,J]$ of the Mean Curvature Flow in $\mathbb{R}^3$ with $2J+1$ ends by stacking the two-plane. This talk is based on joint works with Kapouleas and Shao.