Two-parametric bifurcations of singular points of the normalized Ricci flow on generalized Wallach spaces

Abstract

We study the behavior of a three-dimensional dynamical system with respect to some set S given in 3-dimensional euclidian space. Geometrically such a system arises from the normalized Ricci flow on some class of generalized Wallach spaces that can be described by a real parameter a ∈ (0, 1/2), as for S it represents the set of invariant Riemannian metrics of positive sectional curvature on the Wallach spaces. Establishing that S is bounded by three conic surfaces and regarding the normalized Ricci flow as an abstract dynamical system we find out the character of interrelations between that system and S for all a ∈ (0, 1/2). These results can cover some well-known results, in particular, they can imply that the normalized Ricci flow evolves all generic invariant Riemannian metrics with positive sectional curvature into metrics with mixed sectional curvature on the Wallach spaces corresponding to the cases a ∈ {1/9, 1/8, 1/6} of generalized Wallach spaces.

The evolution of positively curved invariant Riemannian metrics on the Wallach spaces under the Ricci flow

“…In [4,8] we relied on Maple evaluations and our visual observations concerning structural properties of surfaces and curves obtained from ( 6) and ( 7), but justifications of those properties were not included into the text of the papers. Filling these gaps we initiated in [5]. The present paper continues that idea.…”

“…introduced by R. Hamilton in [16] on generalized Wallach spaces. Since then studies related to this topic were continued in [1,3,7,22] concerning classifications of singular (equilibria) points of (4) being Einstein metrics and their bifurcations. The authors of [2,10,11] studied an interesting and quite complicated surface of bifurcations of (4) defined by a symmetric polynomial equation in three variables a 1 , a 2 , a 3 of degree 12.…”

“…The results of Theorem 2 are illustrated in the right panel of Figure 1 for the case a = 1/6, where the curves r 1 , r 2 and r 3 are depicted respectively in magenta, aquamarine and burlywood colors, the curves l 1 , l 2 and l 3 are depicted by cyan colored curves and yellow colored points correspond to singular points of (5).…”

“…It should be noted that we will consider only Riemannian metrics satisfying the unit volume condition x 1 x 2 x 3 = 1 (see [6,8]), and denote by Σ the surface defined by equation V ≡ 1 with V := x 1 x 2 x 3 . In general, surfaces x 1 x 2 x 3 = c, where c > 0, play the significant role for study (5) on generalized Wallach spaces. It is known that any set determined by the equation x 1 x 2 x 3 = c is invariant under (5), moreover x 1 x 2 x 3 = c is its first integral.…”

“…It is known that any set determined by the equation x 1 x 2 x 3 = c is invariant under (5), moreover x 1 x 2 x 3 = c is its first integral. Surfaces x 1 x 2 x 3 = c will also be unstable (or stable) manifolds of ( 5) and contain leading directions of motions of its trajectories (see [5]). Since the right hand sides of ( 5) are all homogeneous, namely…”