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

Abstract

This paper is devoted to the study of the evolution of positively curved metrics on the Wallach spaces SU(3)/T max , Sp(3)/Sp(1) × Sp(1) × Sp(1), and F 4 /Spin(8). We prove that for all Wallach spaces, the normalized Ricci flow evolves all generic invariant Riemannian metrics with positive sectional curvature into metrics with mixed sectional curvature. Moreover, we prove that for the spaces Sp(3)/Sp(1) × Sp(1) × Sp(1) and F 4 /Spin(8), the normalized Ricci flow evolves all generic invariant Riemannian metrics with positive Ricci curvature into metrics with mixed Ricci curvature. We also get similar results for some more general homogeneous spaces.

Ancient solutions of the homogeneous Ricci flow on flag manifolds

“…x 2 for any i = 1, 2. We may omit the term 1) , by using a time reparametrization. Fixed points of the homogeneous Ricci flow on (M = G/K, g) at infinity of M G = R 2 + , can be studied by setting x 2 = 0.…”

“…Hence, when g is an invariant metric, any solution g(t) of (0.1) is also invariant. As a result, in some cases it is possible to solve the system explicitly and proceed to a study of their asymptotic properties, or even specify analytical properties related to different type of singularities and deduce curvature estimates, see [16,28,7,29,18,38,11,13,1,30], and the articles quoted therein. Especially for the non-compact case, note that during the last decade the Ricci flow for homogeneous, or cohomogeneity-one metrics, together with the so-called bracket flow play a key role in the study of the Alekseevsky conjecture, see [47,41,42,43,39,40,12,14].…”

Ricci flow does not preserve positive sectional curvature in dimension four

“…Moreover, it was previously known [Máx14] that Ric > 0 is not preserved in dimension n = 4, even among Kähler metrics, but these examples do not have sec > 0. Although Theorem A does not readily extend to all n > 4, there are examples of homogeneous metrics on flag manifolds of dimensions 6, 12, and 24 with sec > 0 that lose that property when evolved via Ricci flow, see [BW07,CW15,AN16]. A state-of-the-art discussion of Ricci flow invariant curvature conditions can be found in [BCRW19], see also Remark 5.1.…”

Maxima of Curvature Functionals and the Prescribed Ricci Curvature Problem on Homogeneous Spaces

“…A complete classification of such spaces is given in [13,18]. They possess a number of interesting properties, and their geometry has been studied by several authors; see, e.g., [7,1,13,8]. There are several infinite families and 10 isolated examples, excluding products, constructed out of exceptional Lie groups.…”