Minimal hypersurfaces in S⁴(1) by doubling the equatorial S³

For each large enough m we construct by PDE gluing methods a closed embedded minimal hypersurface Mₘ doubling the equatorial three-sphere Seq³ 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⊂ Seq³. 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 Seq² in S³(1).