Ricci curvature and normalized Ricci flow on generalized Wallach spaces

Abstract: We proved that the normalized Ricci flow does not preserve the positivity of Ricci curvature of Riemannian metrics on every generalized Wallach space with $a_1+a_2+a_3\le 1/2$, in particular on the spaces $$\operatorname{SU}(k+l+m)/\operatorname{SU}(k)\times \operatorname{SU}(l) \times \operatorname{SU}(m)$$ and $$\operatorname{Sp}(k+l+m)/\operatorname{Sp}(k)\times \operatorname{Sp}(l) \times \operatorname{Sp}(m)$$ independently on $k,l$ and $m$. The positivity of Ricci curvature is preserved for all original metrics with $\operatorname{Ric}&gt;0$ on generalized Wallach spaces $a_1+a_2+a_3&gt; 1/2$ if the conditions $$4\left(a_j+a_k\right)^2\ge (1-2a_i)(1+2a_i)<sup>{-1}$$ hold for all ${i,j,k}={1,2,3}$. We also established that the spaces $$\operatorname{SO}(k+l+m)/\operatorname{SO}(k)\times \operatorname{SO}(l)\times \operatorname{SO}(m)$$ satisfy the above conditions for $\max{k,l,m}\le 11$, moreover, additional conditions were found to keep $\operatorname{Ric}&gt;0$ in cases when $\max{k,l,m}\le 11$ is violated. Similar questions have also been studied for all other generalized Wallach spaces given in the classification of Yuri\u\i\ Nikonorov.