A family of 4-manifolds with non-negative Ricci curvature and prescribed asymptotic cone

Abstract In this paper, we show that, for any finite subgroup Γ O ⁢ (4) S 3 S^3, there exists a 4-dimensional complete Riemannian manifold (M, g) (M, g) with Ric g ≥ 0 Ric₆ 0 such that the asymptotic cone of (M, g) (M, g) is C ⁢ (S δ 3 / Γ) C (S_^3/) for some δ = δ ⁢ (Γ) > 0 = () >0. This answers a question of Bruè–Pigati–Semola E. Bruè, A. Pigati and D. Semola, Topological regularity and stability of noncollapsed spaces with Ricci curvature bounded below, preprint (2024), https: //arxiv. org/abs/2405. 03839 about the topological obstructions of 4-dimensional non-collapsed tangent cones. Combining this result with the work of Bruè–Pigati–Semola, one can classify the 4-dimensional non-collapsed tangent cone in the topological sense.