Backward Ricci Flow on Locally Homogeneous 4-Manifolds

TL;DR

This paper investigates the long-term behavior of backward Ricci flow on 4-dimensional locally homogeneous manifolds with compact quotients, revealing convergence to sub-Riemannian geometries near singularities.

Contribution

It provides a detailed analysis of the backward Ricci flow on 4-manifolds, classifying their behavior and showing convergence patterns to sub-Riemannian geometries.

Findings

Many classes exhibit similar behavior near singular time

Manifolds often converge to sub-Riemannian geometries after rescaling

The study describes long-term evolution of homogeneous geometries

Abstract

In this paper we study backward Ricci flow of locally homogeneous geometries of $4$-manifolds which admit compact quotients. We describe the long-term behavior of each class and show that many of the classes exhibit the same behavior near the singular time. In most cases, these manifolds converge to a sub-Riemannian geometry after suitable rescaling.