On the Gromov-Hausdorff limits of Tori with Ricci conditions

TL;DR

This work shows that Ricci-lower-bound limits of tori can be non-manifold orbifolds in high dimensions, by constructing a local cone smoothing near $\mathbb{R}P^2$ that tends to $C(\mathbb{S}^3 / Q_8)$ and embedding this into torus topology to obtain non-manifold GH limits. It proves a sharp contrast in dimension four: non-collapsed limits with two-sided Ricci bounds remain topological tori, while Kähler limits in dimension four are holomorphic orbifolds with isolated $\mathbb{R}^4 / Q_8$ singularities. The key technical tool is a smoothing metric $g_{\rho,\phi}$ on the normal bundle that yields nonnegative Ricci curvature and matches a cone at infinity, enabling the global torus constructions and the analysis of singularities. Collectively, the results delineate when Ricci lower bounds preserve manifold topology and when they permit orbifold limits, highlighting the roles of dimension and additional geometric structure (Kähler, polarization).

Abstract

Let $n\geq 4$. In this paper, we construct a sequence of smooth Riemannian metrics $g_i $ on $\mathbb{R}^n$ such that: (1) $g_i = g_{\rm Euc} $ outside the standard Euclidean unit ball $B_1 (0) \subset \mathbb{R}^n $, (2) ${\rm Ric}_{g_i} \geq -Λ$ and $ {\rm diam} \left( B_1 (0) ,g_i \right) \leq D $ for some $Λ,D>0$ independent of $i$, (3) The pointed Gromov-Hausdorff limit of $(\mathbb{R}^n ,g_i) $ is a topological orbifold but not a topological manifold. As a consequence, for $n\geq 4$, we can find a sequence of tori $(T^n , g_i )$ with Ricci lower bound and diameter bound such that the Gromov-Hausdorff limit is not a topological manifold. This answers a question of Bruè-Naber-Semola [arXiv:2307.03824] in the negative. In $4$-dimensional case, we prove that the Gromov-Hausdorff limit of tori with $2$-side Ricci bound and diameter bound is always a topological torus. In the Kähler case, the Gromov-Hausdorff limit of Kähler tori of real dimension $4$ with Ricci lower bound is always a topological orbifold with isolated singularities, and the only type of singularities is $\mathbb{R}^4 / Q_8 $.

Theorems & Definitions (31)

• Theorem 1.1
• Remark
• Remark
• Proposition 1.1
• Proposition 1.2
• Proposition 1.3
• Lemma 2.1
• proof
• Proposition 2.1
• proof