# Differential Geometry, Lie Theory and Low-Dimensional Topology, Melbourne, Australia, 2016

Title The evolution of positively curved invariant metrics on the Wallach spaces under the Ricci flow Yurii Nikonorov (Southern Mathematical Institute VSC RAS) Prof Yurii Nikonorov n/a This talk is based on a recent joint paper with N.A.~Abiev ({\it N.A.~Abiev, Yu.G.~Nikonorov, The evolution of positively curved invariant Riemannian metrics on the Wallach spaces under the Ricci flow, Annals of Global Analysis and Geometry, 50:1 (2016), 65--84}). We studied the evolution of positively curved metrics on the Wallach spaces $SU(3)/T_{\max}$, $Sp(3)/Sp(1)\times Sp(1)\times Sp(1)$, and $F_4/Spin(8)$. We proved that for all Wallach spaces, the Ricci flow evolves all generic invariant Riemannian metrics with positive sectional curvature into metrics with mixed sectional curvature. Moreover, we proved that for the spaces $Sp(3)/Sp(1)\times Sp(1)\times Sp(1)$ and $F_4/Spin(8)$, the Ricci flow evolves all generic invariant Riemannian metrics with positive Ricci curvature into metrics with mixed Ricci curvature. We also obtained similar results for some more general homogeneous spaces.