Minimal hypersurfaces in S^4 (1) by doubling the equatorial S^3

For each large enough m we construct by PDE gluing methods a closed embedded minimal hypersurface Mₘ doubling the equatorial three-sphere S₄ₐ³ in S⁴ (1), with Mₘ containing m² bridges modelled after the three-dimensional catenoid and centered at the points of a square m m lattice L contained in the Clifford torus T² S₄ₐ³. The construction respects the symmetries of the lattice L as a subset of S⁴ (1) and is based on the Linearized Doubling (LD) methodology which was first introduced in the construction of minimal surface doublings of S₄ₐ² in S³ (1).