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

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 ${\mathrm{Ric}}_g\ge 0$, 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 Bruè-Pigati-Semola [arXiv:2405.03839] about the topological obstructions of $4$-dimensional non-collapsed tangent cones. Combining this result with a recent work of Bruè-Pigati-Semola [arXiv:2405.03839], one can classify the $4$-dimensional non-collapsed tangent cone in the topological sense.

26 pages, any comments are welcome!

Cited by in corpus (1)

• The large scale structure of complete $4$-manifolds with nonnegative Ricci curvature and Euclidean volume growth