Refined behavior description of the normalized Ricci flow on homogeneous spaces

TL;DR

This paper investigates the behavior of the normalized Ricci flow on homogeneous spaces, identifying conditions under which Ricci curvature positivity is preserved or lost, with a focus on generalized Wallach spaces.

Contribution

It provides a detailed analysis of Ricci curvature preservation on generalized Wallach spaces under NRF, including infinitely many cases and refinements of previous results.

Findings

Infinitely many GWS preserve Ricci positivity under NRF.

Infinitely many GWS can lose Ricci positivity during NRF.

Refinements to earlier results for specific parameter cases.

Abstract

This article deals with the problems of preserving the Ricci curvature positivity on homogeneous spaces under the normalized Ricci flow (NRF). We found out infinitely many generalized Wallach spaces (GWS) on which the positivity of the Ricci curvature of metrics is preserved when evolved by the NRF. Analogously, the number of GWS is infinite as well, when the positivity of the Ricci curvature can be lost. We also obtain some refinements to our previous results devoted to the case of coincided parameters. A series of examples is discussed.