On the Gromov-Hausdorff limits of Tori with Ricci conditions

Abstract

Let $n\ge 4$. In this paper, we construct a sequence of smooth Riemannian metrics $g_i$ on ${\mathbb{R}}^n$ such that: (1) $g_i=g_{\mathrm{Euc}}$ outside the standard Euclidean unit ball $B_1(0)\subset {\mathbb{R}}^n$, (2) ${\mathrm{Ric}}_{g_i}\ge -\Lambda$ and $\mathrm{diam}\left(B_1(0),g_i\right)\le D$ for some $\Lambda ,D>0$ 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 $n\ge 4$, 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 ${\mathbb{R}}^4/Q_8$.

17 pages, all comments are welcome. Accepted by Amer. J. Math. (Date: 5/12/2025)

Cited by in corpus (1)

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