The Ricci flow approach to homogeneous Einstein metrics on flag manifolds

Abstract

We give the global picture of the normalized Ricci flow on generalized flag manifolds with two or three isotropy summands. The normalized Ricci flow for these spaces descents to a parameter depending system of two or three ordinary differential equations, respectively. We present here the qualitative study of these system's global phase portrait, by using techniques of Dynamical Systems theory. This study allows us to draw conclusions about the existence and the analytical form of invariant Einstein metrics on such manifolds, and seems to offer a better insight to the classification problem of invariant Einstein metrics on compact homogeneous spaces.

arXiv comment17 pages, 2 figures

References in corpus (3)

• Invariant Einstein metrics on flag manifolds with four isotropy summands
• Invariant Einstein metrics on generalized flag manifolds with two isotropy summands
• The Ricci flow of left invariant metrics on full flag manifold SU(3)/T from a dynamical systems point of view

Cited by in corpus (9)

• Non-naturally reductive Einstein metrics on exceptional Lie groups
• Homogeneous Einstein metrics on generalized flag manifolds with five isotropy summands
• The Ricci flow on generalized Wallach spaces
• Flag manifolds, symmetric $\fr{t}$-triples and Einstein metrics
• The classification of homogeneous Einstein metrics on flag manifolds with $b_2(M)=1$
• Modified Ricci flow and asymptotically non-flat spaces
• Homogeneous Einstein metrics on non-Kähler C-spaces
• The projected homogeneous Ricci flow and three-isotropy-summands flag manifolds
• On the embeddability of the homogeneous Ricci flow and its collapses