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 $\operatorname{Ric}>0$ on generalized Wallach spaces $a_1+a_2+a_3> 1/2$ if the conditions $4\left(a_j+a_k\right)^2\ge (1-2a_i)(1+2a_i)^{-1}$ 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 $\operatorname{Ric}>0$ 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.