On the Gromov-Hausdorff limits of Tori with Ricci conditions

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 $.