The Ricci Flow on Some Generalized Wallach Spaces

Abstract

AbstractWe study the asymptotic behavior of the normalized Ricci flow on generalized Wallach spaces that could be considered as a special planar dynamical system. All nonsymmetric 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 \((0,1/2] \times (0,1/2] \times (0,1/2]\). We also consider in detail some important partial cases.KeywordsRiemannian metricEinstein metricGeneralized Wallach spaceRicci flowRicci curvaturePlanar dynamical systemReal algebraic surfaceMathematics Subject Classification (2010):201053C3053C4437C1034C0514P05