https://neckpinch.com/authors/nikolaos-kapouleas/New arXiv papers by Nikolaos Kapouleas, from Neckpinch Tue, 03 Nov 2026 08:41:12 GMT https://neckpinch.com/papers/2405.18283/arxiv:2405.18283Tue, 28 May 2024 00:00:00 GMTNikolaos Kapouleas, Jiahua Zou — For each large enough m∈ℕ we construct by PDE gluing methods a closed embedded smooth minimal hypersurface breveMₘ doubling the equatorial three-sphere Sₑq³ in S⁴(1), with breveMₘ 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. This answers a long-standing question of Yau in the case of S⁴(1) and long-standing questions of Hsiang. Similarly we construct a self-shrinker breveM_(shr,m) of the Mean Curvature Flow in ℝ⁴ doubling the three-dimensional spherical self-shrinker Sₛₕᵣ³⊂ ℝ⁴ with the bridges centered at the points of a square m× m lattice L contained in a Clifford torus T²⊂ Sₛₕᵣ³. Both constructions respect the symmetries of the lattice L as a subset of S⁴(1) or ℝ⁴ and are based on the Linearized Doubling (LD) methodology which was first introduced in the construction of minimal surface doublings of Sₑq² in S³(1). Furthermore breveMₘ converges as m →∞ in the varifold sense to 2Sₑq³, and its volume |breveMₘ| < 2|Sₑq³|.Mean curvature flowSelf-shrinkers & translators3-manifoldsWeak & singular flowsMinimal & CMC surfaces