Mahler type theorem and non-collapsed limits of sub-Riemannian compact Heisenberg manifolds

TL;DR

This paper investigates the limits of sequences of compact Heisenberg manifolds with sub-Riemannian metrics, establishing conditions under which these sequences converge to a limit that is also a Heisenberg manifold, extending to Riemannian cases with Ricci bounds.

Contribution

It proves convergence of non-collapsed sequences of compact Heisenberg manifolds under diameter and measure bounds, including Riemannian cases with Ricci curvature constraints.

Findings

Sequences with bounded diameter and measure have convergent subsequences.

Limits are isometric to compact Heisenberg manifolds.

Results extend to Riemannian manifolds with Ricci curvature bounds.

Abstract

In this paper, we study a non-collapsed Gromov--Hausdorff limit of a sequence of compact Heisenberg manifolds with sub-Riemannian metrics. In the case of strictly sub-Riemannian case, we show that if a sequence has an upper bound of the diameter and a lower bound of Popp's measure, then it has a convergent subsequence in the Gromov--Hausdorff topology, and the limit is isometric to a compact Heisenberg manifold of the same dimension. The same conclusion is also shown for Riemannian case with the additional assumption on Ricci curvature lower bounds.