Stability of tori under lower sectional curvature

Bruè, Elia; Naber, Aaron; Semola, Daniele

doi:10.2140/gt.2024.28.3961

Download this article
	For screen
	For printing

Recent Issues

The Journal

Submission Guidelines

Submission Page

Policies for Authors

Ethics Statement

ISSN 1364-0380 (online)

ISSN 1465-3060 (print)

Author Index

To Appear

Other MSP Journals

Stability of tori under lower sectional curvature

Elia Bruè, Aaron Naber and Daniele Semola

Geometry & Topology 28 (2024) 3961–3972

DOI: 10.2140/gt.2024.28.3961

Abstract

Let $(M_{i}^{n}, g_{i}) \to_{}^{GH} (X, d_{X})$ be a Gromov–Hausdorff converging sequence of Riemannian manifolds with ${S e c}_{g_{i}} \geq - 1$ , $d i a m (M_{i}) \leq D$ , 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 $0 \leq k \leq n$ . 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 ${R i c}_{g_{i}} \geq - 2$ .

Keywords

sectional, curvature, tori, stability

Mathematical Subject Classification

Primary: 53C21

References

Publication

Received: 12 July 2023

Revised: 25 October 2023

Accepted: 1 December 2023

Published: 20 December 2024

Proposed: Leonid Polterovich

Seconded: Dmitri Burago, Gang Tian

Authors

Elia Bruè
Università Bocconi Milan Italy

Aaron Naber
Northwestern University Evanston, IL United States

Daniele Semola
ETH Zurich Switzerland

Open Access made possible by participating institutions via Subscribe to Open.

Volume 28, issue 8 (2024)