Minimal hypersurfaces in ${\mathbb{S}}^4(1)$ by doubling the equatorial ${\mathbb{S}}^3$

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 ${\mathbb{S}}_{\mathrm{e}\mathrm{q}}^3$ in ${\mathbb{S}}^4(1)$, with ${\breve{M}}_m$ containing $m^2$ bridges modelled after the three-dimensional catenoid and centered at the points of a square $m\times m$ lattice $L$ contained in the Clifford torus ${\mathbb{T}}^2\subset {\mathbb{S}}_{\mathrm{e}\mathrm{q}}^3$. This answers a long-standing question of Yau in the case of ${\mathbb{S}}^4(1)$ and long-standing questions of Hsiang. Similarly we construct a self-shrinker ${\breve{M}}_{\mathrm{s}\mathrm{h}\mathrm{r},m}$ of the Mean Curvature Flow in ${\mathbb{R}}^4$ doubling the three-dimensional spherical self-shrinker ${\mathbb{S}}_{\mathrm{s}\mathrm{h}\mathrm{r}}^3\subset {\mathbb{R}}^4$ with the bridges centered at the points of a square $m\times m$ lattice $L$ contained in a Clifford torus ${\mathbb{T}}^2\subset {\mathbb{S}}_{\mathrm{s}\mathrm{h}\mathrm{r}}^3$. Both constructions respect the symmetries of the lattice $L$ as a subset of ${\mathbb{S}}^4(1)$ or ${\mathbb{R}}^4$ and are based on the Linearized Doubling (LD) methodology which was first introduced in the construction of minimal surface doublings of ${\mathbb{S}}_{\mathrm{e}\mathrm{q}}^2$ in ${\mathbb{S}}^3(1)$. Furthermore ${\breve{M}}_m$ converges as $m\to \mathrm{\infty }$ in the varifold sense to $2{\mathbb{S}}_{\mathrm{e}\mathrm{q}}^3$, and its volume $\mathrm{\mid }{\breve{M}}_m\mathrm{\mid }<2\mathrm{\mid }{\mathbb{S}}_{\mathrm{e}\mathrm{q}}^3\mathrm{\mid }$.