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

Abstract

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 $\mathbb{S}_{\mathrm{eq}}^3$ 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 $\mathbb{T}^2\subset \mathbb{S}_{\mathrm{eq}}^3$. 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 ${\breve{M}_{\mathrm{shr},m}}$ of the Mean Curvature Flow in $\mathbb{R}^4$ doubling the three-dimensional spherical self-shrinker $\mathbb{S}_{\mathrm{shr}}^3\subset \mathbb{R}^4$ with the bridges centered at the points of a square $m\times m$ lattice $L$ contained in a Clifford torus $\mathbb{T}^2\subset \mathbb{S}_{\mathrm{shr}}^3$. 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 $\mathbb{S}_{\mathrm{eq}}^2$ in $\mathbb{S}^3(1)$. Furthermore $\breve{M}_m$ converges as $m \to\infty$ in the varifold sense to $2\mathbb{S}_{\mathrm{eq}}^3$, and its volume $|\breve{M}_m| < 2|\mathbb{S}_{\mathrm{eq}}^3|$.