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

TL;DR

This paper investigates how the normalized Ricci flow affects positively curved invariant metrics on Wallach spaces, showing that such metrics generally evolve into ones with mixed curvature properties.

Contribution

It demonstrates that normalized Ricci flow transforms all generic invariant metrics with positive curvature on Wallach spaces into metrics with mixed curvature, extending to some homogeneous spaces.

Findings

Metrics with positive sectional curvature become mixed under Ricci flow.

Metrics with positive Ricci curvature also evolve into mixed curvature metrics.

Results apply to Wallach spaces and some broader homogeneous spaces.

Abstract

This paper is devoted to the study of the evolution of positively curved metrics on the Wallach spaces $SU(3)/T_{\max}$, $Sp(3)/Sp(1)\times Sp(1)\times Sp(1)$, 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 $Sp(3)/Sp(1)\times Sp(1)\times Sp(1)$ 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.