Stability of tori under lower sectional curvature

Bruè, Elia¹ ; Naber, Aaron² ; Semola, Daniele³

¹ Università Bocconi, Milan, Italy
² Northwestern University, Evanston, IL, United States
³ ETH, Zurich, Switzerland

Geometry & topology, Tome 28 (2024) no. 8, pp. 3961-3972.

Résumé

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

Détail

DOI : 10.2140/gt.2024.28.3961

Keywords: sectional, curvature, tori, stability

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