Higher order geometric flows on three dimensional locally homogeneous spaces

TL;DR

This paper investigates second order geometric flows on three-dimensional homogeneous spaces, revealing unique behaviors in scale evolution, singularity formation, and isotropization, with comparisons to Ricci flows.

Contribution

It introduces analysis of second order curvature flows on 3D homogeneous spaces, highlighting differences from Ricci flows and exploring their dynamic features.

Findings

Distinct scale factor behaviors identified

New fixed curves and phase portraits discovered

Insights into singularity approaches and curvature evolution

Abstract

We analyse second order (in Riemann curvature) geometric flows (un-normalised) on locally homogeneous three manifolds and look for specific features through the solutions (analytic whereever possible, otherwise numerical) of the evolution equations. Several novelties appear in the context of scale factor evolution, fixed curves, phase portraits, approaches to singular metrics, isotropisation and curvature scalar evolution. The distinguishing features linked to the presence of the second order term in the flow equation are pointed out. Throughout the article, we compare the results obtained, with the corresponding results for un-normalized Ricci flows.