Cross Curvature Flow on Locally Homogeneous Three-manifolds (II)

TL;DR

This paper analyzes the long-term behavior of positive cross curvature flow on locally homogeneous 3-manifolds, revealing that it often results in Heisenberg type sub-Riemannian geometries, complementing earlier negative flow results.

Contribution

It provides a comprehensive description of the asymptotic behavior of both positive and negative cross curvature flows on these manifolds, highlighting the typical geometric outcomes.

Findings

Positive flow often leads to Heisenberg type geometries

Negative flow behavior previously characterized

Combined results give a full picture of flow dynamics

Abstract

In this paper, we study the positive cross curvature flow on locally homogeneous 3-manifolds. We describe the long time behavior of these flows. We combine this with earlier results concerning the asymptotic behavior of the negative cross curvature flow to describe the two sided behavior of maximal solutions of the cross curvature flow on locally homogeneous 3-manifolds. We show that, typically, the positive cross curvature flow on locally homogeneous 3-manifold produce an Heisenberg type sub-Riemannian geometry.