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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="normal">Γ</m:mi> <m:mo>&lt;</m:mo> <m:mrow> <m:mi>O</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>4</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> \Gamma&lt;O(4) acting freely on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">S</m:mi> <m:mn>3</m:mn> </m:msup> </m:math> \mathbb{S}^{3} , there exists a 4-dimensional complete Riemannian manifold <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>M</m:mi> <m:mo>,</m:mo> <m:mi>g</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (M,g) with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>Ric</m:mi> <m:mi>g</m:mi> </m:msub> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> \mathrm{Ric}_{g}\geq 0 such that the asymptotic cone of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>M</m:mi> <m:mo>,</m:mo> <m:mi>g</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (M,g) is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>C</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msubsup> <m:mi mathvariant="double-struck">S</m:mi> <m:mi>δ</m:mi> <m:mn>3</m:mn> </m:msubsup> <m:mo>/</m:mo> <m:mi mathvariant="normal">Γ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> C(\mathbb{S}_{\delta}^{3}/\Gamma) for some <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>δ</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mi>δ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="normal">Γ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>&gt;</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> \delta=\delta(\Gamma)&gt;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.