Accepted Paper

Inserted: 7 jul 2023
Last Updated: 2 dec 2023

Journal: Geometry & Topology
Year: 2023

Abstract:

Let $(M^n_i, g_i)\stackrel{GH}{\longrightarrow} (X,\dist_X)$ be a Gromov-Hausdorff converging sequence of Riemannian manifolds with ${\rm Sec}_{g_i} \ge -1$, ${\rm diam}\, (M_i)\le D$, and such that the $M^n_i$ are all homeomorphic to tori $T^n$. Then $X$ is homeomorphic to a $k$-dimensional torus $T^k$ for some $0\leq k\leq n$. This answers a question of Petrunin in the affirmative. We show this result is false is the $M^n_i$ are homeomorphic tori which are only assumed to be Alexandrov spaces. When $n=3$, we prove the same tori stability under the weaker condition ${\rm Ric}_{g_i} \ge -2$.

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• torus stability accepted