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Using Algebraic Geometry

We present in this paper a general approach to study the Ricci flow on homogeneous manifolds. Our main tool is a dynamical system defined on a subset $$\mathcal H _{q,n}$$
 of the variety of $$(q+n)$$
 -dimensional Lie algebras, parameterizing the space of all simply connected homogeneous spaces of dimension $$n$$
 with a $$q$$
 -dimensional isotropy, which is proved to be equivalent in a precise sense to the Ricci flow. The approach is useful to better visualize the possible (nonflat) pointed limits of Ricci flow solutions, under diverse rescalings, as well as to determine the type of the possible singularities. Ancient solutions arise naturally from the qualitative analysis of the evolution equation. We develop two examples in detail: a $$2$$
 -parameter subspace of $$\mathcal H _{1,3}$$
 reaching most of $$3$$
 -dimensional geometries, and a $$2$$
 -parameter family in $$\mathcal H _{0,n}$$
 of left-invariant metrics on $$n$$
 -dimensional compact and non-compact semisimple Lie groups.

Ricci flow of homogeneous manifolds

Geometry of compact complex homogeneous spaces with vanishing first Chern class

The study of left-invariant Einstein metrics on compact Lie groups which are naturally reductive was initiated by D’Atri and Ziller (Mem Am Math Soc 18, (215) 1979). In 1996 the second author obtained non-naturally reductive Einstein metrics on the Lie group SU(n) for n ≥ 6, by using a method of Riemannian submersions. In the present work we prove existence of non-naturally reductive Einstein metrics on the compact simple Lie groups SO(n) (n ≥ 11), Sp(n) (n ≥ 3), E6, E7, and E8.

Einstein metrics on compact Lie groups which are not naturally reductive

The Ricci flow approach to homogeneous Einstein metrics on flag manifolds

A generalized flag manifold is a homogeneous space of the form $G/K$, where $K$ is the centralizer of a torus in a compact connected semisimple Lie group $G$. We classify all flag manifolds with four isotropy summands and we study their geometry. We present new $G$-invariant Einstein metrics by solving explicity the Einstein equation. We also examine the isometric problem for these Einstein metrics.

Invariant Einstein metrics on flag manifolds with four isotropy summands

A generalized flag manifold is a homogeneous space of the form G/K, where K is the centralizer of a torus in a compact connected semisimple Lie group G. We classify all flag manifolds with four isotropy summands by the use of $${\mathfrak{t}}$$
 -roots. We present new G-invariant Einstein metrics by solving explicity the Einstein equation. We also examine the isometric problem for these Einstein metrics.

Invariant einstein metrics on flag manifolds with four isotropy summands

Let $M=G/K$ be a generalized flag manifold, that is the adjoint orbit of a compact semisimple Lie group $G$. We use the variational approach to find invariant Einstein metrics for all flag manifolds with two isotropy summands. We also determine the nature of these Einstein metrics as critical points of the scalar curvature functional under fixed volume.

/pdf/invariant-einstein-metrics-on-generalized-flag-manifolds-3ao2m0exbf.pdf

Invariant Einstein metrics on generalized flag manifolds with two isotropy summands

Let M = G/K be a generalized flag manifold, that is the adjoint orbit of a compact semisimple Lie group G. We use the variational approach to find invariant Einstein metrics for all flag manifolds with two isotropy summands. We also determine the nature of these Einstein metrics as critical points of the scalar curvature functional under fixed volume. 2000 Mathematics Subject Classification. Primary 53C25; Secondary 53C30, 22E46

/pdf/invariant-einstein-metrics-on-generalized-flag-manifolds-1iytj1ceg7.pdf

Ioannis Chrysikos

Papers

The Ricci flow approach to homogeneous Einstein metrics on flag manifolds

Invariant Einstein metrics on flag manifolds with four isotropy summands

Invariant einstein metrics on flag manifolds with four isotropy summands

Invariant Einstein metrics on generalized flag manifolds with two isotropy summands

Invariant einstein metrics on generalized flag manifolds with two isotropy summands