The projected homogeneous Ricci flow and its collapses with an application to flag manifolds

On the dynamics of positively curved metrics on SU(3)/T 2 under the homogeneous Ricci flow

“…See, for instance, Theorem 4.1. As expected by the authors of [15], such an approach works well to study the dynamics of some positively curved metrics on certain homogeneous spaces. We state our results before going a bit deeper into the machinery employed.…”

“…The proof of our results is obtained by restricting the dynamics of invariant metrics g = (x, y, z) to the tetrahedron x + y + z = 1 via the machinery in [15], focusing particularly on the case of Riemannian submersion metrics g(t) = (t, t, 1 − 2t). Precisely, we show the family g(t) constitutes invariant solutions to the projected homogeneous Ricci flow (Lemma 4.4), and since it is in hand the complete behavior of the here-treated curvature formulae in terms of t, the proof to our results are straightforwardly obtained.…”

“…On the other hand, the Ricci flow is feasible to handle (due to betterestablished techniques) in the homogeneous setting, [15,25,14,13,26]. Of particular interest is the recent paper [15], substantiated in the following: It consists in appropriately normalizing the Ricci flow to a simplex followed by a time reparametrization to obtain polynomial equations, leading to a projected Ricci flow. This arises from a natural generalization of the standard unit-volume reparametrization of the Ricci flow.…”

“…On the other hand, as it is presented in Section 4.1.2, the Ricci flow system for invariant metrics on W 6 is a system of three nonlinear ODEs (in three variables), which usually would impose more difficulty on handling their dynamics. However, as introduced in [15], we can, for W 6 and other flag manifolds (with three isotropy summands), reduce equivalently the dynamic analyses' to the plane, where dynamical system techniques are easier to employ -this is what is called projected homogeneous Ricci flow.…”

Geodesic orbit spaces in real flag manifolds

“…Exploring the Ricci flow on homogeneous spaces involves tools from the theory of dynamical systems. This approach has been adopted by various works covering different classes of homogeneous spaces, including generalized Wallach spaces [1,23], Stiefel manifolds [23], and complex flag manifolds [13,14,23].…”