A family of $4$-manifolds with nonnegative Ricci curvature and prescribed asymptotic cone

TL;DR

This paper constructs 4-dimensional complete Riemannian manifolds with nonnegative Ricci curvature that have prescribed asymptotic cones, addressing a key topological classification problem for tangent cones in geometric analysis.

Contribution

It demonstrates the existence of 4-manifolds with nonnegative Ricci curvature and specific asymptotic cones, solving a previously open topological obstructions question.

Findings

Existence of manifolds with prescribed asymptotic cones

Classification of 4D non-collapsed tangent cones

Addresses topological obstructions in tangent cone classification

Abstract

In this paper, we show that for any finite subgroup $\Gamma < O(4)$ acting freely on $\mathbb{S}^3$, there exists a $4$-dimensional complete Riemannian manifold $(M,g)$ with ${\rm Ric}_g \geq 0 $, such that the asymptotic cone of $(M,g)$ is $C(\mathbb{S}_\delta^3 /\Gamma )$ for some $\delta = \delta (\Gamma ) >0$. This answers a question of Bru\`e-Pigati-Semola [arXiv:2405.03839] about the topological obstructions of $4$-dimensional non-collapsed tangent cones. Combining this result with a recent work of Bru\`e-Pigati-Semola [arXiv:2405.03839], one can classify the $4$-dimensional non-collapsed tangent cone in the topological sense.