Collapsed limits of compact Heisenberg manifolds with sub-Riemannian metrics

TL;DR

This paper proves that collapsed limits of compact Heisenberg manifolds under sub-Riemannian metrics are flat tori, providing insights into their geometric structure and volume behavior.

Contribution

It establishes that all collapsed Gromov--Hausdorff limits of compact Heisenberg manifolds are isometric to flat tori, revealing their geometric degeneration.

Findings

Collapsed limits are flat tori.

Total measure converges to zero during collapse.

Systolic inequality relates measure exponent to Hausdorff dimension.

Abstract

In this paper, we show that every collapsed Gromov--Hausdorff limit of compact Heisenberg manifolds is isometric to a flat torus. Here we say that a sequence of sub-Riemannian manifolds collapses if their total measure with respect to the Popp's volume or the minimal Popp's volume converges to zero. In the appendix, we give the systolic inequality on sub-Riemannian Heisenberg manifolds, and observe that the exponent of the total measure is equal to the inverse of the Hausdorff dimension.