The Ricci flow on generalized Wallach spaces

TL;DR

This paper analyzes the long-term behavior of the normalized Ricci flow on generalized Wallach spaces, revealing the nature of singularities for most parameter choices within a specific cube.

Contribution

It provides a qualitative classification of singularities of the Ricci flow on generalized Wallach spaces parametrized by three positive numbers.

Findings

Most singular points have classified types of singularities.

The behavior is characterized for parameters in the cube (0,1/2]^3.

The results apply to all non-symmetric generalized Wallach spaces.

Abstract

We consider the asymptotic behavior of the normalized Ricci flow on generalized Wallach spaces that could be considered as special planar dynamical systems. All non symmetric generalized Wallach spaces can be naturally parametrized by three positive numbers $a_1, a_2, a_3$. Our interest is to determine the type of singularity of all singular points of the normalized Ricci flow on all such spaces. Our main result gives a qualitative answer for almost all points $(a_1,a_2,a_3)$ in the cube $(0,1/2]\times (0,1/2]\times (0,1/2]$.