Stability of Tori under Lower Sectional Curvature

TL;DR

The paper proves that sequences of Riemannian tori with bounded lower sectional curvature converge to a torus of lower dimension, confirming a conjecture, and explores stability under weaker curvature conditions and in Alexandrov spaces.

Contribution

It establishes the stability of tori under Gromov-Hausdorff convergence with lower sectional curvature bounds, confirming Petrunin's conjecture and analyzing cases with weaker curvature assumptions.

Findings

Limit space is homeomorphic to a lower-dimensional torus.

Stability fails for Alexandrov tori without Riemannian structure.

3D tori are stable under Ricci curvature bounds.

Abstract

Let $(M^n_i, g_i)\to (X,d_X)$ 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 ${\rm Ric}_{g_i} \ge -2$.