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

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.